Re: The seven step-Mathematical preliminaries
2009/6/17 Torgny Tholerus tor...@dsv.su.se: Bruno Marchal skrev: Torgny, I agree with Quentin. You are just showing that the naive notion of set is inconsistent. Cantor already knew that, and this is exactly what forced people to develop axiomatic theories. So depending on which theory of set you will use, you can or cannot have an universal set (a set of all sets). In typical theories, like ZF and VBG (von Neuman Bernay Gödel) the collection of all sets is not a set. It is not the naive notion of set that is inconsistent. It is the naive *handling* of sets that is inconsistent. This problem has two possible solutions. One possible solution is to deny that it is possible to create the set of all sets. This solution is chosen by ZF and VBG. The second possible solution is to be very careful of the domain of the All quantificator. You are not allowed to substitute an object that is not included in the domain of the quantificator. It is this second solution that I have chosen. What is illegal in the two deductions below, is the substitutions. Because the sets A and B do not belong to the domain of the All quantificator. You can define existence by saying that only that which is incuded in the domain of the All quantificator exists. In that case it is correct to say that the sets A and B do not exist, because they are not included in the domain. But I think this is a too restrictive definition of existence. It is fully possible to talk about the set of all sets. But you must then be *very* careful with what you do with that set. That set is a set, but it does not belong to the set of all sets, it does not belong to itself. It is also a matter of definition; if you define set as the same as belonging to the set of all sets, then the set of all sets is not a set. This is a matter of taste. You can choose whatever you like, but you must be aware of your choice. But if you restrict yourself too much, then your life will be poorer... In NF, some have developed structure with universal sets, and thus universe containing themselves. Abram is interested in such universal sets. And, you can interpret the UD, or the Mandelbrot set as (simple) model for such type of structure. Your argument did not show at all that the set of natural numbers leads to any trouble. Indeed, finitism can be seen as a move toward that set, viewed as an everything, potentially infinite frame (for math, or beyond math, like it happens with comp). The problem of naming (or given a mathematical status) to all sets is akin to the problem of giving a name to God. As Cantor was completely aware of. We are confused on this since we exist. But the natural numbers, have never leads to any confusion, despite we cannot define them. The proof that there is no biggest natural number is illegal, because you are there doing an illegal deduction, you are there doing an illegal substitution, just the same as in the deductions below with the sets A and B. You are there substituting an object that is not part of the domain of the All quatificator. No the proof is based on PA and in PA you do not have an axiom restricting the successor function and as such it is defined in the axiom that you don't have an upper bound limit. The proof is *valid* against the axioms. *You* are doing an illegal deduction by not taking into accound the rules with wich you work. Regards, Quentin -- Torgny Tholerus You argument against the infinity of natural numbers is not valid. You cannot throw out this little infinite by pointing on the problem that some terribly big infinite, like the set of all sets, leads to trouble. That would be like saying that we have to abandon all drugs because the heroin is very dangerous. It is just non valid. Normally, later I will show a series of argument very close to Russell paradoxes, and which will yield, in the comp frame, interesting constraints on what computations are and are not. Bruno On 13 Jun 2009, at 13:26, Torgny Tholerus wrote: Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol- string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you
Re: The seven step-Mathematical preliminaries
Bruno Marchal skrev: Torgny, I agree with Quentin. You are just showing that the naive notion of set is inconsistent. Cantor already knew that, and this is exactly what forced people to develop axiomatic theories. So depending on which theory of set you will use, you can or cannot have an universal set (a set of all sets). In typical theories, like ZF and VBG (von Neuman Bernay Gödel) the collection of all sets is not a set. It is not the naive notion of set that is inconsistent. It is the naive *handling* of sets that is inconsistent. This problem has two possible solutions. One possible solution is to deny that it is possible to create the set of all sets. This solution is chosen by ZF and VBG. The second possible solution is to be very careful of the domain of the All quantificator. You are not allowed to substitute an object that is not included in the domain of the quantificator. It is this second solution that I have chosen. What is illegal in the two deductions below, is the substitutions. Because the sets A and B do not belong to the domain of the All quantificator. You can define existence by saying that only that which is incuded in the domain of the All quantificator exists. In that case it is correct to say that the sets A and B do not exist, because they are not included in the domain. But I think this is a too restrictive definition of existence. It is fully possible to talk about the set of all sets. But you must then be *very* careful with what you do with that set. That set is a set, but it does not belong to the set of all sets, it does not belong to itself. It is also a matter of definition; if you define set as the same as belonging to the set of all sets, then the set of all sets is not a set. This is a matter of taste. You can choose whatever you like, but you must be aware of your choice. But if you restrict yourself too much, then your life will be poorer... In NF, some have developed structure with universal sets, and thus universe containing themselves. Abram is interested in such universal sets. And, you can interpret the UD, or the Mandelbrot set as (simple) model for such type of structure. Your argument did not show at all that the set of natural numbers leads to any trouble. Indeed, finitism can be seen as a move toward that set, viewed as an everything, potentially infinite frame (for math, or beyond math, like it happens with comp). The problem of naming (or given a mathematical status) to all sets is akin to the problem of giving a name to God. As Cantor was completely aware of. We are confused on this since we exist. But the natural numbers, have never leads to any confusion, despite we cannot define them. The proof that there is no biggest natural number is illegal, because you are there doing an illegal deduction, you are there doing an illegal substitution, just the same as in the deductions below with the sets A and B. You are there substituting an object that is not part of the domain of the All quatificator. -- Torgny Tholerus You argument against the infinity of natural numbers is not valid. You cannot throw out this little infinite by pointing on the problem that some terribly big infinite, like the set of all sets, leads to trouble. That would be like saying that we have to abandon all drugs because the heroin is very dangerous. It is just non valid. Normally, later I will show a series of argument very close to Russell paradoxes, and which will yield, in the comp frame, interesting constraints on what computations are and are not. Bruno On 13 Jun 2009, at 13:26, Torgny Tholerus wrote: Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol- string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you accept, that a set can be part of itself, it makes your deduction legal regarding the rules. OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? --- Define the set B of all sets that do not belong to itself as:
Re: The seven step-Mathematical preliminaries
Torgny, I agree with Quentin. You are just showing that the naive notion of set is inconsistent. Cantor already knew that, and this is exactly what forced people to develop axiomatic theories. So depending on which theory of set you will use, you can or cannot have an universal set (a set of all sets). In typical theories, like ZF and VBG (von Neuman Bernay Gödel) the collection of all sets is not a set. In NF, some have developed structure with universal sets, and thus universe containing themselves. Abram is interested in such universal sets. And, you can interpret the UD, or the Mandelbrot set as (simple) model for such type of structure. Your argument did not show at all that the set of natural numbers leads to any trouble. Indeed, finitism can be seen as a move toward that set, viewed as an everything, potentially infinite frame (for math, or beyond math, like it happens with comp). The problem of naming (or given a mathematical status) to all sets is akin to the problem of giving a name to God. As Cantor was completely aware of. We are confused on this since we exist. But the natural numbers, have never leads to any confusion, despite we cannot define them. You argument against the infinity of natural numbers is not valid. You cannot throw out this little infinite by pointing on the problem that some terribly big infinite, like the set of all sets, leads to trouble. That would be like saying that we have to abandon all drugs because the heroin is very dangerous. It is just non valid. Normally, later I will show a series of argument very close to Russell paradoxes, and which will yield, in the comp frame, interesting constraints on what computations are and are not. Bruno On 13 Jun 2009, at 13:26, Torgny Tholerus wrote: Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol- string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you accept, that a set can be part of itself, it makes your deduction legal regarding the rules. OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? --- Define the set B of all sets that do not belong to itself as: For all x holds that x belongs to B if and only if x does not belong to x. This is an general rule saying that for some particular symbol- string x you can always tell if x belongs to B or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for B itself. We can always substitute B for x. Then we will get: B belongs to B if and only if B does not belong to B. --- Quentin, what do you think? Is this deduction legal or illegal? -- Torgny Tholerus http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
On 12 Jun 2009, at 17:16, A. Wolf wrote: We agree then. Yes, it's my fault for creating a semantics argument. I'm usually too busy to even read the list...every once in a while something pops up and I feel obliged to comment even when it's the middle of a conversation. No problem. I actually have some questions for the list members that are relevant to the list content, and this coming week is break. I may have a chance to post them. They're much more on the philosophical side than the mathematical, though. Don't hesitate, Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
On 12 Jun 2009, at 20:31, Jesse Mazer wrote: Even for just an arithmetical realist. (All mathematicians are arithmetical realist, much less are mathematical realist. I am not an arithmetical realist). I assume you meant to write I am not a mathematical realist? Yes. OK, but this leads to a further question. I remember from Penrose's book that he talked about various levels of oracle machines (hypercomputers)--for example, a first order-oracle machine was like a Turing machine but with an added operation that could decide in one step whether any Turing program halts or not, a second-order oracle machine was like a Turing machine but with operations that could decide whether a Turing machine program *or* a first-order oracle machine program halts, and so forth. Hmm... let us say OK (but this could be ambiguous). This gives mainly the arithmetical hierarchy when you start from the oracle for the halting problem. There are relativized hierarchy based on any oracle, and then starting from the halting problem in that oracle. The degrees are structured in a very complex way. I don't know the meaning of the phrases arithmetical hierarchy or relativized hierarchy, is there any simple way of explaining? In any case, the problem I am mainly concerned with is the set of all propositions that would be considered well-formed-formulas (WFFs) in the context of Peano arithmetic (so it would involve arithmetical symbols like + and x as well as logical symbols from first-order logic, and they'd be ordered in such a way that the symbol-string would express a coherent statement about arithmetic that could be either true or false). Is there some way to come up with a rule that would allow you to judge the true or falsity of *every* member of this set of propositions using some kind of sufficiently powerful hypercomputer (presumably a fairly powerful one like the 'omega- order oracle machine' or beyond), just by checking every possible value for the numbers that could be substituted in for variable symbols? That would allow us to make sense of the distinction between truths about arithmetic and statements about arithmetic provable by some axiomatic system like the Peano axioms (the issue Brent Meeker was talking about in the post at http://www.mail-archive.com/everything-list@googlegroups.com/msg16562.html ), without having to worry about the meaning that we assign to arithmetical symbols like the number 2, or about philosophical questions about where our understanding of that meaning comes from. I think you are asking something impossible. The notion of elementary aritmetical truth will always be simpler than the notion you need to define the hypercomputer. Just Gödel's incompleteness theorem justifies in an transparent way the separation between truth and provability in such or such formal theory. The arithmetical hierachy is the one I was describing with the sequence of alternation of quantifiers starting from decidable formula. That hierarchy does indeed described a sequence of hypermachine (because each level posses a Sigma_i or Pi_i completeness notion, which generalize the notion of universality for machine-with-oracle. I will have some opportunity to describe notion even more powerful. But only (with Church thesis) the Sigma_1 universality has an effective universal counterpart, represented by any computer, or universal language interpreter. Instead we'd have a purely formal definition about how to judge the truth-value of WFFs beyond those the Peano axioms can judge, Remember that PA+consistent PA is uncomputably more powerful than PA. And ZF set theory is much much more powerful than PA, ... This is something I will have to explain, at least for AUDA, but in all those discussions we have to distinguih the notion of computability (which is universal and does not depend on the choice of formalism), with the notion of provability which is formalism dependent. And then we can generalize the notion of computability, and even of provability. Humans provability is generally considered as much larger than PA arithmetic. albeit one that cannot actually be put into practice for arbitrary propositions without actually having such a hypercomputer (but for some specific propositions like the Godel statement for the Peano axioms, I think we can come up with an argument for why the hypercomputer should judge this statement 'true' as long as we believe the Peano axioms are consistent, so in this sense defining arithmetical truth in terms of such a hypercomputer is *conceptually* useful). The whole incompleteness phenomena bears on all notion of hypercomputations. I prefer to call hypercomputers Gods or angels, and both humans and machines can accelerate their work by invoking them. The notion of real numbers are based on such
Re: The seven step-Mathematical preliminaries
2009/6/14 Torgny Tholerus tor...@dsv.su.se: Quentin Anciaux skrev: Well it is illegal regarding the rules meaning with these rules set B does not exist as defined. What is it that makes set A to exist, and set B not to exist? What is the (important) differences between the definition of set A and the definition of set B? In both cases you are defining a set by giving a property that all members of the set must fulfill. Yes and one fulfil it according to the given rules the other not. I would add that your excercise is inconsistent from the start, whatever a set is, your argument is contradictory whatever the rules are. Why is the deduction legal for set A, but illegal for set B? There is the same type of deduction in both places, you are just making a substitution for the all quantificator in both cases. That's all the point of puting rules and checking that something is correct or not according to it. 1+1=3 is false according to PA... that doesn't mean you couldn't find a rule or mapping that would render this statement true ***regarding the chosen rules***/ Regards, Quentin -- Torgny Tholerus 2009/6/13 Torgny Tholerus tor...@dsv.su.se: Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you accept, that a set can be part of itself, it makes your deduction legal regarding the rules. OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? --- Define the set B of all sets that do not belong to itself as: For all x holds that x belongs to B if and only if x does not belong to x. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to B or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for B itself. We can always substitute B for x. Then we will get: B belongs to B if and only if B does not belong to B. --- Quentin, what do you think? Is this deduction legal or illegal? -- Torgny Tholerus -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/13 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Fri, 12 Jun 2009 18:40:14 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
It is, as I said above, for me and all other humans to understand what
you are talking about. It is also for to be able to decide what
deductions or conclusions or proofs that are legal or illegal.
Well, most humans who think about mathematics can understand
rule-based definitions like 0 is a whole number, and N is a whole
number if it's equal to some other whole number plus one--you seem to
be the lone exception.
As for being able to decide what deductions or conclusions or proofs
that are legal or illegal, how exactly would writing out all the
members of the universe solve that? For example, I actually write
out all the numbers from 0 to 1,038,712 and say that they are members
of the universe I want to talk about. But if I write out some axioms
used to prove various propositions about these numbers, they are still
going to be in the form of general *rules* with abstract variables
like x and y (where these variables stand for arbitrary numbers in the
set), no? Or do you also insist that instead of writing axioms and
making deductions, we also spell out in advance every proposition that
shall be deemed true? In that case there is no room at all for
mathematicians to make deductions or write proofs, all of math
would just consist of looking at the pre-established list of true
propositions and checking if the proposition in question is on there.
What do you think about the following deduction? Is it legal or illegal?
---
Define the set A of all sets as:
For all x holds that x belongs to A if and only if x is a set.
This is an general rule saying that for some particular symbol-string x
you can always tell if x belongs to A or not. Most humans who think
about mathematics can understand this rule-based definition. This rule
holds for all and every object, without exceptions.
So this rule also holds for A itself. We can always substitute A for
x. Then we will get:
A belongs to A if and only if A is a set.
And we know that A is a set. So from this we can deduce:
A beongs to A.
---
Quentin, what do you think? Is this deduction legal or illegal?
It depends if you allow a set to be part of itselft or not.
If you accept, that a set can be part of itself, it makes your
deduction legal regarding the rules. If you don't then the statement
is illegal regarding the rules (it violates the rule saying that a set
can't contains itself, which means that A in this system is not a set
thus all the reasoning in *that system* is false.
Choosing one rule or the other tells nothing about the rule itself
unless you can find a contradiction by choosing one or the other.
Regards,
Quentin
But I can't see why a set as I understand it cannot be part of
itself... {1,2,3} is included in {1,2,3} is true, what is the exact
problem with that statement ? (written differently all elements of the
set A are elements of the set B === A is included in B, here as A and
B are the same A is included in A.
--
Torgny Tholerus
--
All those moments will be lost in time, like tears in rain.
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Re: The seven step-Mathematical preliminaries
Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you accept, that a set can be part of itself, it makes your deduction legal regarding the rules. OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? --- Define the set B of all sets that do not belong to itself as: For all x holds that x belongs to B if and only if x does not belong to x. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to B or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for B itself. We can always substitute B for x. Then we will get: B belongs to B if and only if B does not belong to B. --- Quentin, what do you think? Is this deduction legal or illegal? -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Well it is illegal regarding the rules meaning with these rules set B does not exist as defined. 2009/6/13 Torgny Tholerus tor...@dsv.su.se: Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you accept, that a set can be part of itself, it makes your deduction legal regarding the rules. OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? --- Define the set B of all sets that do not belong to itself as: For all x holds that x belongs to B if and only if x does not belong to x. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to B or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for B itself. We can always substitute B for x. Then we will get: B belongs to B if and only if B does not belong to B. --- Quentin, what do you think? Is this deduction legal or illegal? -- Torgny Tholerus -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Sat, 13 Jun 2009 11:05:22 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Fri, 12 Jun 2009 18:40:14 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries It is, as I said above, for me and all other humans to understand what you are talking about. It is also for to be able to decide what deductions or conclusions or proofs that are legal or illegal. Well, most humans who think about mathematics can understand rule-based definitions like 0 is a whole number, and N is a whole number if it's equal to some other whole number plus one--you seem to be the lone exception. As for being able to decide what deductions or conclusions or proofs that are legal or illegal, how exactly would writing out all the members of the universe solve that? For example, I actually write out all the numbers from 0 to 1,038,712 and say that they are members of the universe I want to talk about. But if I write out some axioms used to prove various propositions about these numbers, they are still going to be in the form of general *rules* with abstract variables like x and y (where these variables stand for arbitrary numbers in the set), no? Or do you also insist that instead of writing axioms and making deductions, we also spell out in advance every proposition that shall be deemed true? In that case there is no room at all for mathematicians to make deductions or write proofs, all of math would just consist of looking at the pre-established list of true propositions and checking if the proposition in question is on there. What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. It's well known that if you allow sets to contain themselves, and allow arbitrary rules for what a given set can contain, then you can get contradictions like Russell's paradox (the set of all sets which do not contain themselves). But what relevance does this have to arithmetic? Are you afraid the basic Peano axioms might lead to two propositions which can be derived in finite time from the axioms but which are mutually contradictory? If so I don't see how allowing only a finite collection of numbers actually helps--like I said in an earlier post, the number of propositions that can be proved about a finite set of numbers can still be infinite. I suppose it might be possible to make it finite by disallowing propositions which are created merely by connecting other propositions with the AND or OR logical operators, but it's still the case that if your largest whole number BIGGEST is supposed to be at least as large as some numbers humans have already conceived--say, as large as 10^100--then there is no way we could actually write out all possible propositions about these numbers that follow from some Peano-like axiom system to check manually that no two propositions contradicted each other (do you want to try to calculate 10^100 + A and A + 10^100 for every possible value of A smaller than 10^100 to verify explicitly that they are equal in every case?) So, it seems that unless you want to make your universe of numbers *very* small, you have to rely on some sort of mental model of arithmetic to be confident that you won't get contradictions from the axioms you start from, just like how people are confident in the non-contradictoriness of the Peano axioms based on their mental model of counting discrete objects like marbles (see my comments in the last paragraph of the post at http://www.mail-archive.com/everything-list@googlegroups.com/msg16564.html ). Jesse --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
There, they call arithmetic soundness what me (and many logician) call soundness, when they refer to theories about numbers. Like Mendelson I prefer to use the term logically valid, to what you call soundness. I may have misstated myself, but the wiki article you pointed me to agrees with what I tried to say: A logical system is sound if every provable statement is valid. Validity is not the same as soundness. There are valid arguments that are unsound. For example, if I say x is not equal to x, therefore there are no more than five natural numbers, this is a valid (i.e., logically true) argument. But it's also an unsound argument, because there is no interpretation where x is not equal to x. What you're calling soundness I would call omega-consistent, but I see from the article that this is sometimes called arithmetical soundness. The word true alone has no meaning. It refers always to a model, or to a collection of models. One could make the same argument about the symbol = not having any meaning outside of a model, but true has a standard meaning in logic, one that is often used interchangeably with valid (a stronger property). The general true means true under any interpretation. Oh, you mean a definition of natural number such that the model would be finite in scope. This is non sense for me. Pace Torgny. Nonsense for me too, apart from the philisophical musings. Well, there is just no categorical first order definition of the finite sets of natural numbers. And second order definition, assumes the notion of infinite set. I'm not sure what you mean here. Of course there is no categorical first-order theory of N. Anna --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
On 11 Jun 2009, at 21:43, Jesse Mazer wrote: Countably infinite does not mean recursively countably infinite. This is something which I will explain in the seventh step thread. There is theorem by Kleene which links Post-Turing degrees of unsolvability with the shape of arithmetical formula. With P denoting decidable predicates (Sigma_0) we have ExP(x) Sigma_1 (mechanical) AxP(x) Pi_1 ExAyP(x,y) Sigma_2 AxEyP(x,y) Pi_2 etc. Ah, that makes sense--I hadn't thought of combining multiple universal quantifiers in that way, but obviously you can do so and get a meaningful statement about arithmetic that for a mathematical realist should be either true or false. Even for just an arithmetical realist. (All mathematicians are arithmetical realist, much less are mathematical realist. I am not an arithmetical realist). Of course some exotic philosopher could pretend they are not arithmetical realist, but this is near non sense for me. OK, but this leads to a further question. I remember from Penrose's book that he talked about various levels of oracle machines (hypercomputers)--for example, a first order-oracle machine was like a Turing machine but with an added operation that could decide in one step whether any Turing program halts or not, a second-order oracle machine was like a Turing machine but with operations that could decide whether a Turing machine program *or* a first-order oracle machine program halts, and so forth. Hmm... let us say OK (but this could be ambiguous). This gives mainly the arithmetical hierarchy when you start from the oracle for the halting problem. There are relativized hierarchy based on any oracle, and then starting from the halting problem in that oracle. The degrees are structured in a very complex way. You can go even past finite-order oracle machines into oracle machines for higher ordinals too... This leads to the hyperarithmetical hierarchy and/or analytical hierarchy, where you consider formula with variables for functions or sets. There are many non trivial theorems which relate those notions (and open problems, but I have not follow the recent developments). Imo, the best book on that subject is still the book by Rogers. for example, an omega-order oracle machine can tell you whether any finite-order oracle machine halts, an omega-plus-one-order oracle machine can tell you whether any finite-order oracle machine halts *or* whether an omega-order oracle machine halts, and so forth. So I assume from what you're saying above that even an omega-order oracle machine would not be able to decide the truth value of every proposition about arithmetic just by checking cases...if that's right, what would the propositions it can't decide look like? It can't just follow the pattern you showed above of adding more universal quantifiers, since it has to be a proposition of finite length. Indeed, but variable can represent infinite object, like in analysis. From the point of view of computability, infinite set of natural number, or functions from N to N, can play the role of the real numbers. I've also read that countable ordinals themselves can be classed as either computable or noncomputable (which makes sense since I'm pretty sure you can come up with a formalism where every possible countable ordinal is associated with a countable symbol-string, although the same ordinal might have multiple valid ways of expressing it as a symbol string since order doesn't matter in sets, so this doesn't help with the problem of whether the cardinality of the set of all distinct countable ordinals is the same as the cardinality of the set of all distinct real numbers, i.e. all distinct countable symbol-strings). So, there is a first countable- but-noncomputable ordinal, written as omega_1^CK (where _1 refers to a subscript and ^CK refers to a superscript), Yes. And CK is for Church and Kleene. You can see omega_1^CK as the least non recursive ordinal. Omega_1 (aleph_1) is the least non countable ordinal. Of course omega_1^CK is much smaller than Omega_1. And with the continuum hypothesis Omega_1 is 2^aleph_0. which means we should also have a notion of an omega_1^CK-order oracle machine. You are quick here! There are more than one way to make this precise. Would there be finite propositions about arithmetic that even this fantastical device could not decide? This can depend of which path you will follow going through the constructive ordinals, and yes some path define fantastical device capable of answering all arithmetical questions. Obviously, this correspond to anything but machines! But yes, with comp those objects makes non sense form the third person point of view, but still are needed to figure out the first person points of view. Can all propositions about arbitrary *real*
Re: The seven step-Mathematical preliminaries
Le 12-juin-09, à 09:31, Bruno Marchal a écrit : On 11 Jun 2009, at 21:43, Jesse Mazer wrote: Ah, that makes sense--I hadn't thought of combining multiple universal quantifiers in that way, but obviously you can do so and get a meaningful statement about arithmetic that for a mathematical realist should be either true or false. Even for just an arithmetical realist. (All mathematicians are arithmetical realist, much less are mathematical realist. I am not an arithmetical realist). Raaah... Sorry. Of course I am an arithmetical realist. I am not a *mathematical* realist (still less a physical realist). With the month of June I have a lot of works to do, and I tend to do it simultaneously. The result is an augmentation of mistakes! I hope you have learned to automatically correct them. Sorry. Please, ask in case of remaining doubt. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Le 12-juin-09, à 08:28, A. Wolf a écrit : There, they call arithmetic soundness what me (and many logician) call soundness, when they refer to theories about numbers. Like Mendelson I prefer to use the term logically valid, to what you call soundness. I may have misstated myself, but the wiki article you pointed me to agrees with what I tried to say: That wiki article is not so good. A logical system is sound if every provable statement is valid. ... meaning true in all models of the theory. OK. Like a tautology, for example P OR NOT P is true is all models (those with P true, and those with P false). A propositional calculus is sound if it proves only tautologies (true in all models), and complete if it proves all tautologies, that is: all the propositions which are true in all models. The intuitive idea is that a reasoning is valid if its truth status does not depend on the way we interpret it. Validity is not the same as soundness. Logicians from different fields use terms in different ways. In provability logic and in recursion theory, soundness means often arithmetical soundness. For example, in recursion theory, theory will be said Sigma_2 sound when all Sigma_2 propositions proved in the theory are true ... in the usual model (N,+,*). There are valid arguments that are unsound. For example, if I say x is not equal to x, therefore there are no more than five natural numbers, this is a valid (i.e., logically true) argument. But it's also an unsound argument, because there is no interpretation where x is not equal to x. Here most, if not all logicians, would disagree. Both in classical and intuitionistic logic, To deduce any proposition from a falsity is always a valid argument. Nobody will say that an argument is non valid because its premise are absurd. Except in the relevance logic field. Well, they will say that the reasoning is not ... relevant. What you're calling soundness I would call omega-consistent, but I see from the article that this is sometimes called arithmetical soundness. Soundness is a semantical notion. By the Tarski phenomenon such a notion cannot be even just defined or expressed in the theory. That is why Gödel makes the terrible effort for not using such a notion, which was considered a bit controversial in those days. Omega-consistency, like consistency, can be defined in a purely syntactical way, and is much weaker than soundness. The word true alone has no meaning. It refers always to a model, or to a collection of models. One could make the same argument about the symbol = not having any meaning outside of a model, but true has a standard meaning in logic, one that is often used interchangeably with valid (a stronger property). The general true means true under any interpretation. That is validity for me. But let us not debate on vocabulary, especially before making a bit more logic. Oh, you mean a definition of natural number such that the model would be finite in scope. This is non sense for me. Pace Torgny. Nonsense for me too, apart from the philisophical musings. OK. Well, there is just no categorical first order definition of the finite sets of natural numbers. And second order definition, assumes the notion of infinite set. I'm not sure what you mean here. Of course there is no categorical first-order theory of N. We agree then. For the others, a theory is categorical if all its model are isomorphic. In a sense, such a theory succeeds in capturing completely its semantics. By well know theorems, such theories are very rare, and in fact, when effective, very poor and very exceptional. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Logicians from different fields use terms in different ways. In provability logic and in recursion theory, soundness means often arithmetical soundness. I understand. Part of the reason for my particular viewpoint: there's a group of professors at the college I work at who are working on bottom-up provability of computer programs under specific constraints. Soundness for them means the proof systems they use never prove a program is correct (meaning, it meets formal, mathematically-written specifications) when it's actually incorrect, provided the constraints are applied correctly. When the constraints are not applied correctly, it's acceptable for the system to prove things that aren't correct, and they still consider this sound for their purposes. We agree then. Yes, it's my fault for creating a semantics argument. I'm usually too busy to even read the list...every once in a while something pops up and I feel obliged to comment even when it's the middle of a conversation. I actually have some questions for the list members that are relevant to the list content, and this coming week is break. I may have a chance to post them. They're much more on the philosophical side than the mathematical, though. Anna --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev: Date: Wed, 10 Jun 2009 09:18:10 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Tue, 9 Jun 2009 18:38:23 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries For you to be able to use the word all, you must define the domain of that word. If you do not define the domain, then it will be impossible for me and all other humans to understand what you are talking about. OK, so how do you say I should define this type of universe? Unless you are demanding that I actually give you a list which spells out every symbol-string that qualifies as a member, can't I simply provide an abstract *rule* that would allow someone to determine in principle if a particular symbol-string they are given qualifies? Or do you have a third alternative besides spelling out every member or giving an abstract rule? You have to spell out every member. Where does this have to come from? Again, is it something you have a philosophical or logical definition for, or is it just your aesthetic preference? It is, as I said above, for me and all other humans to understand what you are talking about. It is also for to be able to decide what deductions or conclusions or proofs that are legal or illegal. It has nothing to do with my aesthetic preference. Because in a *rule* you are (implicitely) using this type of universe, and you will then get a circular definition. A good rule (as opposed to a 'bad' rule like 'the set of all sets that do not contain themselves') gives a perfectly well-defined criteria for what is contained in the universe, such that no one will ever have cause to be unsure about whether some particular symbol-string they're given at belongs in this universe. It's only circular if you say in advance that there is something problematic about rules which define infinite universes, but again this just seems like your aesthetic preference and not something you have given any philosophical/logical justification for. What do you mean by some particular symbol-string? I suppose that you mean by this is: If you take any particular symbol-string from this universe, then no one will ever have cause to be unsure about whether this symbol-string belongs in this universe. So you are defining this universe by supposing that you have this universe to start with. Is that not a typical circular definition? -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Fri, 12 Jun 2009 18:40:14 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Wed, 10 Jun 2009 09:18:10 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Tue, 9 Jun 2009 18:38:23 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries For you to be able to use the word all, you must define the domain of that word. If you do not define the domain, then it will be impossible for me and all other humans to understand what you are talking about. OK, so how do you say I should define this type of universe? Unless you are demanding that I actually give you a list which spells out every symbol-string that qualifies as a member, can't I simply provide an abstract *rule* that would allow someone to determine in principle if a particular symbol-string they are given qualifies? Or do you have a third alternative besides spelling out every member or giving an abstract rule? You have to spell out every member. Where does this have to come from? Again, is it something you have a philosophical or logical definition for, or is it just your aesthetic preference? It is, as I said above, for me and all other humans to understand what you are talking about. It is also for to be able to decide what deductions or conclusions or proofs that are legal or illegal. It has nothing to do with my aesthetic preference. Well, most humans who think about mathematics can understand rule-based definitions like 0 is a whole number, and N is a whole number if it's equal to some other whole number plus one--you seem to be the lone exception. What's more, I kind of think you're playing dumb here, because I bet *you* would have little problem with a rule-based definition of a finite set that didn't actually spell out every member, like 0 is a member of the set, and N is in the set if it's equal to some other member of the set plus one, *unless* the 'other member of the set' is equal to 1,038,712 in which case no members of the set are larger than that. Here, can't you understand that the set includes every whole number from 0 to 1,038,712 without my having to write out every member? And the mental process that allows you to decide whether some string of symbols (say, 1692) would qualify as a member of that set is exactly the same as the mental process that would allow you to decide whether some string of symbols would qualify as a member of the whole numbers which have no upper limit. As for being able to decide what deductions or conclusions or proofs that are legal or illegal, how exactly would writing out all the members of the universe solve that? For example, I actually write out all the numbers from 0 to 1,038,712 and say that they are members of the universe I want to talk about. But if I write out some axioms used to prove various propositions about these numbers, they are still going to be in the form of general *rules* with abstract variables like x and y (where these variables stand for arbitrary numbers in the set), no? Or do you also insist that instead of writing axioms and making deductions, we also spell out in advance every proposition that shall be deemed true? In that case there is no room at all for mathematicians to make deductions or write proofs, all of math would just consist of looking at the pre-established list of true propositions and checking if the proposition in question is on there. Because in a *rule* you are (implicitely) using this type of universe, and you will then get a circular definition. A good rule (as opposed to a 'bad' rule like 'the set of all sets that do not contain themselves') gives a perfectly well-defined criteria for what is contained in the universe, such that no one will ever have cause to be unsure about whether some particular symbol-string they're given at belongs in this universe. It's only circular if you say in advance that there is something problematic about rules which define infinite universes, but again this just seems like your aesthetic preference and not something you have given any philosophical/logical justification for. What do you mean by some particular symbol-string? I suppose that you mean by this is: If you take any particular symbol-string from this universe, then no one will ever have cause to be unsure about whether this symbol-string belongs in this universe. So you are defining this universe by supposing that you have this universe to start with. Is that not a typical circular definition? No, I'm saying take some particular collection of symbols like 0,1,2,3,4,5,6,7,8,9, then any finite ordered group of them
RE: The seven step-Mathematical preliminaries
From: marc...@ulb.ac.be To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Date: Fri, 12 Jun 2009 09:31:46 +0200 On 11 Jun 2009, at 21:43, Jesse Mazer wrote: Countably infinite does not mean recursively countably infinite. This is something which I will explain in the seventh step thread. There is theorem by Kleene which links Post-Turing degrees of unsolvability with the shape of arithmetical formula. With P denoting decidable predicates (Sigma_0) we have ExP(x) Sigma_1 (mechanical) AxP(x) Pi_1ExAyP(x,y) Sigma_2AxEyP(x,y) Pi_2etc. Ah, that makes sense--I hadn't thought of combining multiple universal quantifiers in that way, but obviously you can do so and get a meaningful statement about arithmetic that for a mathematical realist should be either true or false. Even for just an arithmetical realist. (All mathematicians are arithmetical realist, much less are mathematical realist. I am not an arithmetical realist). I assume you meant to write I am not a mathematical realist? OK, but this leads to a further question. I remember from Penrose's book that he talked about various levels of oracle machines (hypercomputers)--for example, a first order-oracle machine was like a Turing machine but with an added operation that could decide in one step whether any Turing program halts or not, a second-order oracle machine was like a Turing machine but with operations that could decide whether a Turing machine program *or* a first-order oracle machine program halts, and so forth. Hmm... let us say OK (but this could be ambiguous). This gives mainly the arithmetical hierarchy when you start from the oracle for the halting problem. There are relativized hierarchy based on any oracle, and then starting from the halting problem in that oracle. The degrees are structured in a very complex way. I don't know the meaning of the phrases arithmetical hierarchy or relativized hierarchy, is there any simple way of explaining? In any case, the problem I am mainly concerned with is the set of all propositions that would be considered well-formed-formulas (WFFs) in the context of Peano arithmetic (so it would involve arithmetical symbols like + and x as well as logical symbols from first-order logic, and they'd be ordered in such a way that the symbol-string would express a coherent statement about arithmetic that could be either true or false). Is there some way to come up with a rule that would allow you to judge the true or falsity of *every* member of this set of propositions using some kind of sufficiently powerful hypercomputer (presumably a fairly powerful one like the 'omega-order oracle machine' or beyond), just by checking every possible value for the numbers that could be substituted in for variable symbols? That would allow us to make sense of the distinction between truths about arithmetic and statements about arithmetic provable by some axiomatic system like the Peano axioms (the issue Brent Meeker was talking about in the post at http://www.mail-archive.com/everything-list@googlegroups.com/msg16562.html ), without having to worry about the meaning that we assign to arithmetical symbols like the number 2, or about philosophical questions about where our understanding of that meaning comes from. Instead we'd have a purely formal definition about how to judge the truth-value of WFFs beyond those the Peano axioms can judge, albeit one that cannot actually be put into practice for arbitrary propositions without actually having such a hypercomputer (but for some specific propositions like the Godel statement for the Peano axioms, I think we can come up with an argument for why the hypercomputer should judge this statement 'true' as long as we believe the Peano axioms are consistent, so in this sense defining arithmetical truth in terms of such a hypercomputer is *conceptually* useful). You can go even past finite-order oracle machines into oracle machines for higher ordinals too... This leads to the hyperarithmetical hierarchy and/or analytical hierarchy, where you consider formula with variables for functions or sets. There are many non trivial theorems which relate those notions (and open problems, but I have not follow the recent developments). Imo, the best book on that subject is still the book by Rogers. Again, I'm only interested here in the type of propositions that would be judged WFFs in the context of the Peano axioms, and I think in this case the variables only refer to particular numbers, right? Or is it possible to write a WFF in this context where the variables refer to functions or sets? Like you I am an arithmetical realist but not necessarily a realist about arbitrary sets. I think it may be problematic to talk about sets that are so big that most of their numbers have no finite description, as must be true of any uncountable set. If there is no *rule* which maps countable ordinals
RE: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 20:00, Brent Meeker wrote: Bruno Marchal wrote: On 10 Jun 2009, at 02:20, Brent Meeker wrote: I think Godel's imcompleteness theorem already implies that there must be non-unique extensions, (e.g. maybe you can add an axiom either that there are infinitely many pairs of primes differing by two or the negative of that). That would seem to be a reductio against the existence of a hypercomputer that could decide these propositions by inspection. Not at all. Gödel's theorem implies that there must be non-unique *consistent* extensions. But there is only one sound extension. The unsound consistent extensions, somehow, does no more talk about natural numbers. OK. But ISTM that statement implies that we are relying on an intuitive notion as our conception of natural numbers, rather than a formal definition. You are right. We have to rely on our intuition. After Gödel we know that even our use of formal system has to be based on our intuition of the natural number, and we don't have any fixed and complete formalization of the natural numbers. I guess I don't understand unsound in this context. Unsound means false in the structure (N, +, *). We can define this formally in a theory which is richer than PA, like ZF set theory. Of course we can't define sound for ZF. Intuition just cannot be avoided. Today we can understand how machine can develop intuition, despite this cannot be formalized. Typical example: take the proposition that PA is inconsistant. By Gödel's second incompletenss theorem, we have that PA+PA is inconsistent is a consistent extension of PA. But it is not a sound one. It affirms the existence of a number which is a Gödel number of a proof of 0=1. But such a number is not a usual number at all. Suppose, for example, that the twin primes conjecture is undecidable in PA. Are you saying that either PA+TP or PA+~TP must be unsound? Yes. That is why, unlike in Set Theory, nobody seriously doubt about the excluded middle principle in the structure (N,+,*). And what exactly does unsound mean? It really means false in the structure (N,+,*). Does it have a formal definition or does it just mean violating our intuition about numbers? As I said, you can formalize the notion of soundness in Set Theory. But this adds nothing, except that it shows that the notion of soundness has the same level of complexity that usual analytical or topological set theoretical notions. So you can also say that unsound means violation of our intuitive understanding of what the structure (N,+,*) consists in. We cannot formalize in any absolute way that understanding, but we can formalize it in richer theories used everyday by mathematicians. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 20:17, Jesse Mazer wrote: From: marc...@ulb.ac.be To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Date: Wed, 10 Jun 2009 18:03:26 +0200 On 10 Jun 2009, at 01:50, Jesse Mazer wrote: Such an hypercomputer is just what Turing called an oracle. And the haslting oracle is very low in the hierarchy of possible oracles. And Turing results is that even a transfinite ladder of more and more powerful oracles that you can add on Peano Arithmetic, will not give you a complete theory. Hypercomputing by constructive extension of PA, with more and more powerful oracles does not help to overcome incompleteness, unless you add non constructive ordinal extension of hypercomputation. This is the obeject of the study of the degrees of unsolvability, originated by Emil Post. Interesting, thanks. But I find it hard to imagine what kind of finite proposition about natural numbers could not be checked simply by plugging in every possible value for whatever variables appear in the proposition...certainly as long as the number of variables appearing in the proposition is finite, the number of possible ways of substituting specific values for those variables is countably infinite and a hypercomputer should be able to check every case in a finite time. Countably infinite does not mean recursively countably infinite. This is something which I will explain in the seventh step thread. There is theorem by Kleene which links Post-Turing degrees of unsolvability with the shape of arithmetical formula. With P denoting decidable predicates (Sigma_0) we have ExP(x) Sigma_1 (mechanical) AxP(x) Pi_1 ExAyP(x,y) Sigma_2 AxEyP(x,y) Pi_2 etc. This will defined countably infinite set which are more and more complex, and which needs more and more non-mechanical procedures. You can intuitively understand, perhaps, that to be the coded of an halting procedure (Sigma_1) needs less hypercomputation than to be the coded of an everywhere defined procedure, which is Sigma_2. Does what you're saying imply you can you have a proposition which somehow implicitly involves an infinite number of distinct variables even though it doesn't actually write them all out? The complexity grows up even when you restrict yourself to a finite number of variables. By the theorem of Kleene, the complexity comes from the alternation of the quantifiers. Can all propositions about arbitrary *real* numbers (which are of course uncountably infinite) be translated into equivalent propositions about whole numbers in arithmetic? Here you are jumping from arithmetical truth to analytical truth. In principle analytical truth extends the whole arithmetical hierarchy. So the correct answer is no. But this is for arbitrary analytical truth. By a sort of miracle, the analytical truth that we met in the everyday practice of analysis can be translated in arithmetical proposition. A well known example is the Riemann's hypothesis which is equivalent to a Pi_1 arithmetical proposition. I have personally stopped to believe in the relevance of analytical truth in the ontology. Epistemologically, it is not difficult to build arithmetical relation such that you need analytical devices to solve them. A bit like higher cardinal in set theory can provide light in combinatorial problems in braids and knots theory. Or am I taking the wrong approach here, and the reason a hypercomputer can't decide every proposition about arithmetic unrelated to the issue of how many distinct variables can appear in a proposition? It is related to the number of variables, but the hierarchy grows up without necessitating to go beyond finite number of variables. The interesting story about the degree of complexity of hypermachines happens between the recursively countable and the less and less recursively countable, and they are all captured by formula with finite number of variables. I hope I will be able to put some light on this in the seventh step thread, or in some possible AUDA thread in the future. The quantified guardian angel, that is the modal logic G* extended with the quantifier, is already undecidable even in the presence of an oracle for the whole arithmetical truth. Even a GOD or hypermachine capable of answering all Sigma_i and Pi_i questions can still not answer general provability question bearing on a machine. The arithmetical second Plotinian God, that is Plato's NOUS, or intellect, is already beyond the reach of the first Plotinian God (the ONE, or arithmetical truth). No machine can make a complete theory of what machine can and cannot do. When the complexity of machine go above the treshold of universality, they are faced with an intrinsically huge complexity. Machines can understand themselves only very partially. They can progress by transforming
Re: The seven step-Mathematical preliminaries
As I said, you can formalize the notion of soundness in Set Theory. But
this adds nothing, except that it shows that the notion of soundness has
the same level of complexity that usual analytical or topological set
theoretical notions. So you can also say that unsound means violation
of our intuitive understanding of what the structure (N,+,*) consists in.
We cannot formalize in any absolute way that understanding, but we can
formalize it in richer theories used everyday by mathematicians.
You're using soundness in a different sense than I'm familiar with.
Soundness is a property of logical systems that states in this proof
system, provable implies true. Godel's Completeness Theorem shows there
exists a system of logic (first-order logic, specifically) that has this
soundness property. In other words, nothing for which an exact and complete
proof in first-order logic exists, is false.
Soundness is particularly important to logicians because if a system is
unsound, any proofs made with that system are essentially meaningless.
There are limits to what you can do with higher-order logical systems
because of this.
I think what you're bickering over isn't the soundness of the system. I
think it's the selection of the label natural number, which is a
completely arbitrary label. Any definition for natural number which is
finite in scope refers to a different concept than the one we mean when we
say natural number. Any finite subset of N is less useful for
mathematical proofs (and in some cases, much harder to define--not all
subsets of N are definable in the structure {N: +, *}, after all) than the
whole shebang, which is why we immediately prefer the infinite definition.
Anna
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Re: The seven step-Mathematical preliminaries
On 11 Jun 2009, at 14:48, A. Wolf wrote:
As I said, you can formalize the notion of soundness in Set
Theory. But
this adds nothing, except that it shows that the notion of
soundness has
the same level of complexity that usual analytical or topological
set
theoretical notions. So you can also say that unsound means
violation
of our intuitive understanding of what the structure (N,+,*)
consists in.
We cannot formalize in any absolute way that understanding, but
we can
formalize it in richer theories used everyday by mathematicians.
You're using soundness in a different sense than I'm familiar with.
I am indeed not using the term soundness like it is used in
soundness and completeness theorem, like for first order predicate
logic.
I use it, like many provability logician to mean true in the
standard (usual) model of arithmetic.
See
http://en.wikipedia.org/wiki/Soundness
There, they call arithmetic soundness what me (and many logician) call
soundness, when they refer to theories about numbers.
Like Mendelson I prefer to use the term logically valid, to what you
call soundness.
Should not be a problem in this list, given that we don't use the
notion of models, nor of logical validity. I refer very rarely to
Gödel's completness, and when I do so, I do it in the form a theory
has a model iff it is consistent (this can be proved to be the case
for first order theory).
Soundness is a property of logical systems that states in this proof
system, provable implies true. Godel's Completeness Theorem shows
there
exists a system of logic (first-order logic, specifically) that has
this
soundness property. In other words, nothing for which an exact and
complete
proof in first-order logic exists, is false.
Soundness is particularly important to logicians because if a system
is
unsound, any proofs made with that system are essentially meaningless.
There are limits to what you can do with higher-order logical systems
because of this.
I am not sure I follow you. You mean by true, I guess true in, or
satisfied by, all models, or false in any models. A theory is sound
if what is provable in the theory is satisfied by (true in) all models
of the theory.
A deduction A = B is sound, or logically valid, if all models which
satisfy A satisfy B.
The word true alone has no meaning. It refers always to a model, or
to a collection of models.
I think what you're bickering over isn't the soundness of the system.
It is the arithmetical soundness.
I
think it's the selection of the label natural number, which is a
completely arbitrary label.
Nooo, come on.
Any definition for natural number which is
finite in scope refers to a different concept than the one we mean
when we
say natural number.
I don't see what you mean here. Robinson Arithmetic, which is a finite
theory, can be see as a definition of the usual natural numbers, but
like any definitions, finite or infinite (but then recursively
axiomatisable), it has non standard models satisfying the definition.
Oh, you mean a definition of natural number such that the model would
be finite in scope. This is non sense for me. Pace Torgny.
Any finite subset of N is less useful for
mathematical proofs (and in some cases, much harder to define--not all
subsets of N are definable in the structure {N: +, *}, after all)
than the
whole shebang, which is why we immediately prefer the infinite
definition.
Well, there is just no categorical first order definition of the
finite sets of natural numbers. And second order definition, assumes
the notion of infinite set.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
A. Wolf wrote:
As I said, you can formalize the notion of soundness in Set Theory. But
this adds nothing, except that it shows that the notion of soundness has
the same level of complexity that usual analytical or topological set
theoretical notions. So you can also say that unsound means violation
of our intuitive understanding of what the structure (N,+,*) consists in.
We cannot formalize in any absolute way that understanding, but we can
formalize it in richer theories used everyday by mathematicians.
You're using soundness in a different sense than I'm familiar with.
Soundness is a property of logical systems that states in this proof
system, provable implies true. Godel's Completeness Theorem shows there
exists a system of logic (first-order logic, specifically) that has this
soundness property. In other words, nothing for which an exact and complete
proof in first-order logic exists, is false.
I'm not sure I understand this. True and false are just arbitrary
attributes of propositions in logic. I read you last sentence above as saying:
Given premises, which I assume true, then any inference from them using
first-order logic will be true. But that just means I will not be able to
infer a contradiction (=false). In other words, first-order logic is
consistent.
Of course if I start with contradictory premises I will be able construct a
proof in first order logic that proves X and not-X which is false.
Brent
Soundness is particularly important to logicians because if a system is
unsound, any proofs made with that system are essentially meaningless.
There are limits to what you can do with higher-order logical systems
because of this.
I think what you're bickering over isn't the soundness of the system. I
think it's the selection of the label natural number, which is a
completely arbitrary label. Any definition for natural number which is
finite in scope refers to a different concept than the one we mean when we
say natural number. Any finite subset of N is less useful for
mathematical proofs (and in some cases, much harder to define--not all
subsets of N are definable in the structure {N: +, *}, after all) than the
whole shebang, which is why we immediately prefer the infinite definition.
Anna
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Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev: Date: Tue, 9 Jun 2009 18:38:23 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries For you to be able to use the word all, you must define the domain of that word. If you do not define the domain, then it will be impossible for me and all other humans to understand what you are talking about. OK, so how do you say I should define this type of universe? Unless you are demanding that I actually give you a list which spells out every symbol-string that qualifies as a member, can't I simply provide an abstract *rule* that would allow someone to determine in principle if a particular symbol-string they are given qualifies? Or do you have a third alternative besides spelling out every member or giving an abstract rule? You have to spell out every member. Because in a *rule* you are (implicitely) using this type of universe, and you will then get a circular definition. When you say that *every* number have a successor, you are presupposing that you already know what *every* means. -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Torgny, your par. 1: I like your including all universes into *UNIVERSE* if you talk about it. WE, here in this universe think about them. No contact, no lead, just our mental efforts. It all occurs in our prceived reality by thinking about more. your par.2: domain is tricky. I like to write about 'totality' vs 'models i.e. the identified cuts of it for our interest (other lists, other topics) and a smart fellow (NZ) replied: your 'totality' IS a model. You identified it as 'all' (we can imagine) - which is not all that can, or cannot exist. Possible, or impossible in our present views. BTW to 'understand' what somebody talks about is also tricky: we can only translate the 3rd pers. communication into our 1st pers. mindset so what we understand is not (necessarily) what the other said. Or wanted to say. Mindset is individual, no two persons can match in genetic origin (DNA, input of lineage, circumstances in gestational development, plus plus plus), AND the accumulated (personal) experience-material as applied to the individual life-history and emotional responses. Duo si faciunt idem, non est idem valid in ideation as well. I once wrote a sci-fi with an intelligent alien society where the communication consisted of direct transfer of ideas. There was NO discussion. Respectfully John Mikes On Tue, Jun 9, 2009 at 12:38 PM, Torgny Tholerus tor...@dsv.su.se wrote: Jesse Mazer skrev: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. What's a value set? And why do you say we must associate it in this way? Do you have a philosophical argument for this must, or is it just an edict that reflects your personal aesthetic preferences? Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. *I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? When I talk about universe here, I do not mean our physical universe. What I mean is something that can be called everything. It includes all objects in our physical universe, as well as all symbols and all words and all numbers and all sets and all other universes. It includes everything you can use the word all about. For you to be able to use the word all, you must define the domain of that word. If you do not define the domain, then it will be impossible for me and all other humans to understand what you are talking about. -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 01:50, Jesse Mazer wrote: Isn't this based on the idea that there should be an objective truth about every well-formed proposition about the natural numbers even if the Peano axioms cannot decide the truth about all propositions? I think that the statements that cannot be proved are disproved would all be ones of the type for all numbers with property X, Y is true or there exists a number (or some finite group of numbers) with property X (i.e. propositions using either the 'for all' or 'there exists' universal quantifiers in logic, with variables representing specific numbers or groups of numbers). So to believe these statements are objectively true basically means there would be a unique way to extend our judgment of the truth-values of propositions from the judgments already given by the Peano axioms, in such a way that if we could flip through all the infinite propositions judged true by the Peano axioms, we would *not* find an example of a proposition like for this specific number N with property X, Y is false (which would disprove the 'for all' proposition above), and likewise we would not find that for every possible number (or group of numbers) N, the Peano axioms proved a proposition like number N does not have property X (which would disprove the 'there exists' proposition above). We can't actual flip through an infinite number of propositions in a finite time of course, but if we had a hypercomputer that could do so (which is equivalent to the notion of a hypercomputer that can decide in finite time if any given Turing program halts or not), Such an hypercomputer is just what Turing called an oracle. And the haslting oracle is very low in the hierarchy of possible oracles. And Turing results is that even a transfinite ladder of more and more powerful oracles that you can add on Peano Arithmetic, will not give you a complete theory. Hypercomputing by constructive extension of PA, with more and more powerful oracles does not help to overcome incompleteness, unless you add non constructive ordinal extension of hypercomputation. This is the obeject of the study of the degrees of unsolvability, originated by Emil Post. Arithmetical truth is big. No notion of hypercomputing can really help. Yet, the notion of arithmetical truth is well understood by everybody, and is easily definable (as opposed to effectively decidable or computable) in usual set theory. That is why logician have no problem with the notion of standrd model of Peano Arithmetic, for example. then I think we'd have a well-defined notion of how to program it to decide the truth of every for all or there exists proposition in a way that's compatible with the propositions already proved by the Peano axioms. Hypercomputation will not help. Unless you go to the higher non constructive transfinite. But of course, in that case you are using a theory much more powerful than peano Arithmetic and its extension by constructive ordinal. You have to already believe in the notion of truth on numbers to do that. If I'm right about that, it would lead naturally to the idea of something like a unique consistent extension of the Peano axioms (not a real technical term, I just made up this phrase, but unless there's an error in my reasoning I imagine mathematicians have some analogous notion...maybe Bruno knows?) Just go to set theory. Arithmetical truth, or standard model of PA, can play that role. It is not effective (constructive) but it is well defined. Mathematicians used such notions everyday. If you belive in the excluded middle principle on closed arithmetical sentences, you are using implictly such notions. which assigns truth values to all the well-formed propositions that are undecidable by the Peano axioms themselves. You can do that in set theory. Of course, this is not an effective way to do it, but we know, by Godel, that completeness can never be given in any effective way. Set theory can define the standard model (your unique extension) of PA, but it is not a constructive object. In set theory, few object are constructive. So this would be a natural way of understanding the idea of truths about the natural numbers that are not decidable by the Peano axioms. After Godel, truth, even on numbers can be well defined, in richer theory, but have to be non effective, non mechanical. It is not a reason to doubt about the truth of the arithmetical propositions. On the contrary it shows that such truth kicks back and refiute all effective definition we could belive in about that whole truth. Of course even if the notion of a unique consistent extension of certain types of axiomatic systems is well-defined, it would only make sense for axiomatic systems that are consistent in the first place. I guess in judging the question of the
Re: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 02:20, Brent Meeker wrote: I think Godel's imcompleteness theorem already implies that there must be non-unique extensions, (e.g. maybe you can add an axiom either that there are infinitely many pairs of primes differing by two or the negative of that). That would seem to be a reductio against the existence of a hypercomputer that could decide these propositions by inspection. Not at all. Gödel's theorem implies that there must be non-unique *consistent* extensions. But there is only one sound extension. The unsound consistent extensions, somehow, does no more talk about natural numbers. Typical example: take the proposition that PA is inconsistant. By Gödel's second incompletenss theorem, we have that PA+PA is inconsistent is a consistent extension of PA. But it is not a sound one. It affirms the existence of a number which is a Gödel number of a proof of 0=1. But such a number is not a usual number at all. An oracle for the whole arithmetical truth is well defined in set theory, even if it is a non effective object. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 04:14, Jesse Mazer wrote: I think I remember reading in one of Roger Penrose's books that there is a difference between an ordinary consistency condition (which just means that no two propositions explicitly contradict each other) and omega-consistency--see http://en.wikipedia.org/wiki/Omega-consistent_theory . I can't quite follow the details, but I'm guessing the condition means (or at least includes) something like the idea that if you have a statement of the form there exists a number (or set of numbers) with property X then there must actually be some other proposition describing a particular number (or set of numbers) does in fact have this property. The fact that you can add either a Godel statement or its negation to the Peano axioms without creating a contradiction (as long as the Peano axioms are not inconsistent) may not mean you can add either one and still have an omega-consistent theory; if that's true, would there be a unique omega-consistent way to set the truth value of all well-formed propositions about arithmetic which are undecidable by the Peano axioms? Again, Bruno might know... The notion of omega-consistency is a red-herring. The notion exists only for technical reason. Gödel did not succeed in proving the undecidability of its Gödel-sentences without using it, but this will be done succesfully by Rosser. Smullyan introduced better notion than omega-consistency, like his notion of stability, but personaly I prefer to use the (non effective, ok) notion of soundness, and use the notion of stability only latter in more advanced course. The notion of arithmetical soundness was not well seen at the time of Gödel, due to historical circumstances. That's all. But the answer is no. There are non unique omega-consistent extension of PA. Omega-consistency is just a bit more powerful than consistency for proviong undecidability, but Rosser has been able to replace omega-consistency by consistency in the proof of the existence of undecidable statements. Would Gödel have seen Rosser point before Rosser, the notion of omega-consistency could have not appeared at all. I will probably come back on stability, consistency and soundness when we arrive at the AUDA part. This is not for tomorrow. I can give references, well see my URL. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Bruno Marchal wrote: On 10 Jun 2009, at 02:20, Brent Meeker wrote: I think Godel's imcompleteness theorem already implies that there must be non-unique extensions, (e.g. maybe you can add an axiom either that there are infinitely many pairs of primes differing by two or the negative of that). That would seem to be a reductio against the existence of a hypercomputer that could decide these propositions by inspection. Not at all. Gödel's theorem implies that there must be non-unique *consistent* extensions. But there is only one sound extension. The unsound consistent extensions, somehow, does no more talk about natural numbers. OK. But ISTM that statement implies that we are relying on an intuitive notion as our conception of natural numbers, rather than a formal definition. I guess I don't understand unsound in this context. Typical example: take the proposition that PA is inconsistant. By Gödel's second incompletenss theorem, we have that PA+PA is inconsistent is a consistent extension of PA. But it is not a sound one. It affirms the existence of a number which is a Gödel number of a proof of 0=1. But such a number is not a usual number at all. Suppose, for example, that the twin primes conjecture is undecidable in PA. Are you saying that either PA+TP or PA+~TP must be unsound? And what exactly does unsound mean? Does it have a formal definition or does it just mean violating our intuition about numbers? Brent An oracle for the whole arithmetical truth is well defined in set theory, even if it is a non effective object. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Wed, 10 Jun 2009 09:18:10 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Tue, 9 Jun 2009 18:38:23 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries For you to be able to use the word all, you must define the domain of that word. If you do not define the domain, then it will be impossible for me and all other humans to understand what you are talking about. OK, so how do you say I should define this type of universe? Unless you are demanding that I actually give you a list which spells out every symbol-string that qualifies as a member, can't I simply provide an abstract *rule* that would allow someone to determine in principle if a particular symbol-string they are given qualifies? Or do you have a third alternative besides spelling out every member or giving an abstract rule? You have to spell out every member. Where does this have to come from? Again, is it something you have a philosophical or logical definition for, or is it just your aesthetic preference? Because in a *rule* you are (implicitely) using this type of universe, and you will then get a circular definition. A good rule (as opposed to a 'bad' rule like 'the set of all sets that do not contain themselves') gives a perfectly well-defined criteria for what is contained in the universe, such that no one will ever have cause to be unsure about whether some particular symbol-string they're given at belongs in this universe. It's only circular if you say in advance that there is something problematic about rules which define infinite universes, but again this just seems like your aesthetic preference and not something you have given any philosophical/logical justification for. Jesse --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
From: marc...@ulb.ac.be To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Date: Wed, 10 Jun 2009 18:03:26 +0200 On 10 Jun 2009, at 01:50, Jesse Mazer wrote: Isn't this based on the idea that there should be an objective truth about every well-formed proposition about the natural numbers even if the Peano axioms cannot decide the truth about all propositions? I think that the statements that cannot be proved are disproved would all be ones of the type for all numbers with property X, Y is true or there exists a number (or some finite group of numbers) with property X (i.e. propositions using either the 'for all' or 'there exists' universal quantifiers in logic, with variables representing specific numbers or groups of numbers). So to believe these statements are objectively true basically means there would be a unique way to extend our judgment of the truth-values of propositions from the judgments already given by the Peano axioms, in such a way that if we could flip through all the infinite propositions judged true by the Peano axioms, we would *not* find an example of a proposition like for this specific number N with property X, Y is false (which would disprove the 'for all' proposition above), and likewise we would not find that for every possible number (or group of numbers) N, the Peano axioms proved a proposition like number N does not have property X (which would disprove the 'there exists' proposition above). We can't actual flip through an infinite number of propositions in a finite time of course, but if we had a hypercomputer that could do so (which is equivalent to the notion of a hypercomputer that can decide in finite time if any given Turing program halts or not), Such an hypercomputer is just what Turing called an oracle. And the haslting oracle is very low in the hierarchy of possible oracles.And Turing results is that even a transfinite ladder of more and more powerful oracles that you can add on Peano Arithmetic, will not give you a complete theory. Hypercomputing by constructive extension of PA, with more and more powerful oracles does not help to overcome incompleteness, unless you add non constructive ordinal extension of hypercomputation.This is the obeject of the study of the degrees of unsolvability, originated by Emil Post. Interesting, thanks. But I find it hard to imagine what kind of finite proposition about natural numbers could not be checked simply by plugging in every possible value for whatever variables appear in the proposition...certainly as long as the number of variables appearing in the proposition is finite, the number of possible ways of substituting specific values for those variables is countably infinite and a hypercomputer should be able to check every case in a finite time. Does what you're saying imply you can you have a proposition which somehow implicitly involves an infinite number of distinct variables even though it doesn't actually write them all out? Can all propositions about arbitrary *real* numbers (which are of course uncountably infinite) be translated into equivalent propositions about whole numbers in arithmetic? Or am I taking the wrong approach here, and the reason a hypercomputer can't decide every proposition about arithmetic unrelated to the issue of how many distinct variables can appear in a proposition? Jesse --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Of course, Torgny stops, in the UD Argument, at step 0. He disbelieves
classical computationalism.
The yes doctor is made senseless; because he is a zombie, and
Church thesis becomes senseless, because he is ultrafinitist, and
Church thesis concerns functions from N to N, or from N to 2, and NXN
to N, ... It concerns those computable and non computable objects. N=
{0, 1, 2, 3, ...}.
But yes, we need N, and its structure (N,+,x). We cannot prove that N
exists. But we can postulate its existence, give it a recursive name,
and generate and develop more and more simple and powerful theories
about it and its structure. Usual math use N, and its images all the
times. Only a philosopher can be paid to doubt N. A good thing!
Without N, no universal machine, no universal person.
And no Mandelbrot Set M is available for an ultrafinitist, given the
bijection between N and the little Mandelbrots (those the M set is
made of!).
Here
http://www.youtube.com/watch?v=1l9N5a0nxuQfeature=channel
A beautiful illustration that the M set summarizes its histories, 2
times, 4 times, 8 times 16 times, 32 times ... around its little
Mandelbrot sets, (or around its histories ...). In the zoom here, a
feature of the history is going near the tail of a little Mandelbrot
set, and both the music and image coloring (different in the zoom in
and the zoom out) illustrates that Hopf bifurcation where the
neighborhoods are multiplied by two, iteratively, and with an
accelerating frequence, so that the limit (of 2^n) gives a little
mandelbrot set (or ...).
Bruno
On 10 Jun 2009, at 18:24, Bruno Marchal wrote:
On 10 Jun 2009, at 02:20, Brent Meeker wrote:
So we believe in the consistency of Peano's arithmetic because we
have a
physical model.
Why physical? And do we have a physical model? I would say we belive
in the consistency (and soundness) of PA because we have a model of
PA, the well known structure (N, 0, +, *).
If comp is true, there is no physical model at all. (But this is not
something on which I want to insist for now).
Bruno
http://iridia.ulb.ac.be/~marchal/
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. What's a value set? And why do you say we must associate it in this way? Do you have a philosophical argument for this must, or is it just an edict that reflects your personal aesthetic preferences? Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. *I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? When I talk about universe here, I do not mean our physical universe. What I mean is something that can be called everything. It includes all objects in our physical universe, as well as all symbols and all words and all numbers and all sets and all other universes. It includes everything you can use the word all about. For you to be able to use the word all, you must define the domain of that word. If you do not define the domain, then it will be impossible for me and all other humans to understand what you are talking about. -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/9 Torgny Tholerus tor...@dsv.su.se: Jesse Mazer skrev: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. What's a value set? And why do you say we must associate it in this way? Do you have a philosophical argument for this must, or is it just an edict that reflects your personal aesthetic preferences? Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. *I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? When I talk about universe here, I do not mean our physical universe. What I mean is something that can be called everything. It includes all objects in our physical universe, as well as all symbols and all words and all numbers and all sets and all other universes. It includes everything you can use the word all about. It includes all set, but no all set as it N includes all natural number but not all natural number... excuse-me but this is non-sense. Either N exists and has an infinite number of member and is incompatible with an ultrafinitist view or N does not exists because obviously N cannot be defined in an ultra-finitist way, any set that contains a finite number of natural number (and still you haven't defined what it is in an ultrafinitist way) are not the set N. Also any operation involving two number (addition/multiplication) can yield as result a number which has the same property as the departing number (being a natural number) but is not natural number... Also induction and inference cannot work in such a context. For you to be able to use the word all, you must define the domain of that word. If you do not define the domain, then it will be impossible for me and all other humans to understand what you are talking about. Well you are the first and only human I know who don't understand all as everybody else does. Quentin Anciaux -- Torgny Tholerus -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
My philosophical argument is about the mening of the word all. To be
able to use that word, you must associate it with a value set.
What's a value set? And why do you say we must associate it in
this way? Do you have a philosophical argument for this must, or is
it just an edict that reflects your personal aesthetic preferences?
Mostly that set is all objects in the universe, and if you stay
inside the
universe, there is no problems.
*I* certainly don't define numbers in terms of any specific mapping
between numbers and objects in the universe, it seems like a rather
strange notion--shall we have arguments over whether the number 113485
should be associated with this specific shoelace or this specific
kangaroo?
When I talk about universe here, I do not mean our physical universe.
What I mean is something that can be called everything. It includes
all objects in our physical universe, as well as all symbols and all
words and all numbers and all sets and all other universes. It includes
everything you can use the word all about.
It includes all set, but no all set as it N includes all natural
number but not all natural number... excuse-me but this is non-sense.
Either N exists and has an infinite number of member and is
incompatible with an ultrafinitist view or N does not exists because
obviously N cannot be defined in an ultra-finitist way,
That's not obvious to me. You're assuming that N exists apart from
whatever definition of it is given and that it is the infinite set
described by the Peano axioms or equivalent. But that's begging the
question of whether a finite set of numbers that we would call natural
numbers can be defined. To avoid begging the question we need some
definition of natural that doesn't a priori assume the set is finite
or infinite; something like, A set of numbers adequate to do all
arithmetic we'll ever need (unfortunately not very definite). The
problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
except for the case n=N where S{N}=0 and choose sufficiently large N it
might satisfy the natural criteria.
Brent
any set that
contains a finite number of natural number (and still you haven't
defined what it is in an ultrafinitist way) are not the set N.
Also any operation involving two number (addition/multiplication) can
yield as result a number which has the same property as the departing
number (being a natural number) but is not natural number... Also
induction and inference cannot work in such a context.
For you to be able to use the word all, you must define the domain
of that word. If you do not define the domain, then it will be
impossible for me and all other humans to understand what you are
talking about.
Well you are the first and only human I know who don't understand
all as everybody else does.
Quentin Anciaux
--
Torgny Tholerus
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
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Re: The seven step-Mathematical preliminaries
You have to explain why the exception is needed in the first place...
The rule is true until the rule is not true anymore, ok but you have
to explain for what sufficiently large N the successor function would
yield next 0 and why or to add that N and that exception to the
successor function as axiom, if not you can't avoid N+1. But torgny
doesn't evacuate N+1, merely it allows his set to grows undefinitelly
as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
, is a natural number but not part of the set of natural number, this
is non-sense, assuming your special successor rule BIGGEST+1 simply
does not exists at all.
I can understand this overflow successor function for a finite data
type or a real machine registe but not for N. The successor function
is simple, if you want it to have an exception at biggest you should
justify it.
Regards,
Quentin
2009/6/9 Brent Meeker meeke...@dslextreme.com:
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
My philosophical argument is about the mening of the word all. To be
able to use that word, you must associate it with a value set.
What's a value set? And why do you say we must associate it in
this way? Do you have a philosophical argument for this must, or is
it just an edict that reflects your personal aesthetic preferences?
Mostly that set is all objects in the universe, and if you stay
inside the
universe, there is no problems.
*I* certainly don't define numbers in terms of any specific mapping
between numbers and objects in the universe, it seems like a rather
strange notion--shall we have arguments over whether the number 113485
should be associated with this specific shoelace or this specific
kangaroo?
When I talk about universe here, I do not mean our physical universe.
What I mean is something that can be called everything. It includes
all objects in our physical universe, as well as all symbols and all
words and all numbers and all sets and all other universes. It includes
everything you can use the word all about.
It includes all set, but no all set as it N includes all natural
number but not all natural number... excuse-me but this is non-sense.
Either N exists and has an infinite number of member and is
incompatible with an ultrafinitist view or N does not exists because
obviously N cannot be defined in an ultra-finitist way,
That's not obvious to me. You're assuming that N exists apart from
whatever definition of it is given and that it is the infinite set
described by the Peano axioms or equivalent. But that's begging the
question of whether a finite set of numbers that we would call natural
numbers can be defined. To avoid begging the question we need some
definition of natural that doesn't a priori assume the set is finite
or infinite; something like, A set of numbers adequate to do all
arithmetic we'll ever need (unfortunately not very definite). The
problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
except for the case n=N where S{N}=0 and choose sufficiently large N it
might satisfy the natural criteria.
Brent
any set that
contains a finite number of natural number (and still you haven't
defined what it is in an ultrafinitist way) are not the set N.
Also any operation involving two number (addition/multiplication) can
yield as result a number which has the same property as the departing
number (being a natural number) but is not natural number... Also
induction and inference cannot work in such a context.
For you to be able to use the word all, you must define the domain
of that word. If you do not define the domain, then it will be
impossible for me and all other humans to understand what you are
talking about.
Well you are the first and only human I know who don't understand
all as everybody else does.
Quentin Anciaux
--
Torgny Tholerus
--
All those moments will be lost in time, like tears in rain.
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
Everything List group.
To post to this group, send email to everything-list@googlegroups.com
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com
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-~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
I think that resorting to calling the biggest natural number BIGGEST,
rather than specifying exactly what that number is, is a tell-tale sign
that the ultrafinitist knows that any specification for BIGGEST will
immediately reveal that it is not the biggest because one could always
add one more.
Quentin Anciaux wrote:
You have to explain why the exception is needed in the first place...
The rule is true until the rule is not true anymore, ok but you have
to explain for what sufficiently large N the successor function would
yield next 0 and why or to add that N and that exception to the
successor function as axiom, if not you can't avoid N+1. But torgny
doesn't evacuate N+1, merely it allows his set to grows undefinitelly
as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
, is a natural number but not part of the set of natural number, this
is non-sense, assuming your special successor rule BIGGEST+1 simply
does not exists at all.
I can understand this overflow successor function for a finite data
type or a real machine registe but not for N. The successor function
is simple, if you want it to have an exception at biggest you should
justify it.
Regards,
Quentin
2009/6/9 Brent Meeker meeke...@dslextreme.com:
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
My philosophical argument is about the mening of the word all. To be
able to use that word, you must associate it with a value set.
What's a value set? And why do you say we must associate it in
this way? Do you have a philosophical argument for this must, or is
it just an edict that reflects your personal aesthetic preferences?
Mostly that set is all objects in the universe, and if you stay
inside the
universe, there is no problems.
*I* certainly don't define numbers in terms of any specific mapping
between numbers and objects in the universe, it seems like a rather
strange notion--shall we have arguments over whether the number 113485
should be associated with this specific shoelace or this specific
kangaroo?
When I talk about universe here, I do not mean our physical universe.
What I mean is something that can be called everything. It includes
all objects in our physical universe, as well as all symbols and all
words and all numbers and all sets and all other universes. It includes
everything you can use the word all about.
It includes all set, but no all set as it N includes all natural
number but not all natural number... excuse-me but this is non-sense.
Either N exists and has an infinite number of member and is
incompatible with an ultrafinitist view or N does not exists because
obviously N cannot be defined in an ultra-finitist way,
That's not obvious to me. You're assuming that N exists apart from
whatever definition of it is given and that it is the infinite set
described by the Peano axioms or equivalent. But that's begging the
question of whether a finite set of numbers that we would call natural
numbers can be defined. To avoid begging the question we need some
definition of natural that doesn't a priori assume the set is finite
or infinite; something like, A set of numbers adequate to do all
arithmetic we'll ever need (unfortunately not very definite). The
problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
except for the case n=N where S{N}=0 and choose sufficiently large N it
might satisfy the natural criteria.
Brent
any set that
contains a finite number of natural number (and still you haven't
defined what it is in an ultrafinitist way) are not the set N.
Also any operation involving two number (addition/multiplication) can
yield as result a number which has the same property as the departing
number (being a natural number) but is not natural number... Also
induction and inference cannot work in such a context.
For you to be able to use the word all, you must define the domain
of that word. If you do not define the domain, then it will be
impossible for me and all other humans to understand what you are
talking about.
Well you are the first and only human I know who don't understand
all as everybody else does.
Quentin Anciaux
--
Torgny Tholerus
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
Everything List group.
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Re: The seven step-Mathematical preliminaries
Let me correct...
Assuming your special successor rule BIGGEST+1 simply is 0 and is well
defined and *is part* of the previously defined set of natural number
(defined as 0,...,BIGGEST) unlike what Torgny argues.
Regards,
Quentin
2009/6/9 Quentin Anciaux allco...@gmail.com:
You have to explain why the exception is needed in the first place...
The rule is true until the rule is not true anymore, ok but you have
to explain for what sufficiently large N the successor function would
yield next 0 and why or to add that N and that exception to the
successor function as axiom, if not you can't avoid N+1. But torgny
doesn't evacuate N+1, merely it allows his set to grows undefinitelly
as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
, is a natural number but not part of the set of natural number, this
is non-sense, assuming your special successor rule BIGGEST+1 simply
does not exists at all.
I can understand this overflow successor function for a finite data
type or a real machine registe but not for N. The successor function
is simple, if you want it to have an exception at biggest you should
justify it.
Regards,
Quentin
2009/6/9 Brent Meeker meeke...@dslextreme.com:
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
My philosophical argument is about the mening of the word all. To be
able to use that word, you must associate it with a value set.
What's a value set? And why do you say we must associate it in
this way? Do you have a philosophical argument for this must, or is
it just an edict that reflects your personal aesthetic preferences?
Mostly that set is all objects in the universe, and if you stay
inside the
universe, there is no problems.
*I* certainly don't define numbers in terms of any specific mapping
between numbers and objects in the universe, it seems like a rather
strange notion--shall we have arguments over whether the number 113485
should be associated with this specific shoelace or this specific
kangaroo?
When I talk about universe here, I do not mean our physical universe.
What I mean is something that can be called everything. It includes
all objects in our physical universe, as well as all symbols and all
words and all numbers and all sets and all other universes. It includes
everything you can use the word all about.
It includes all set, but no all set as it N includes all natural
number but not all natural number... excuse-me but this is non-sense.
Either N exists and has an infinite number of member and is
incompatible with an ultrafinitist view or N does not exists because
obviously N cannot be defined in an ultra-finitist way,
That's not obvious to me. You're assuming that N exists apart from
whatever definition of it is given and that it is the infinite set
described by the Peano axioms or equivalent. But that's begging the
question of whether a finite set of numbers that we would call natural
numbers can be defined. To avoid begging the question we need some
definition of natural that doesn't a priori assume the set is finite
or infinite; something like, A set of numbers adequate to do all
arithmetic we'll ever need (unfortunately not very definite). The
problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
except for the case n=N where S{N}=0 and choose sufficiently large N it
might satisfy the natural criteria.
Brent
any set that
contains a finite number of natural number (and still you haven't
defined what it is in an ultrafinitist way) are not the set N.
Also any operation involving two number (addition/multiplication) can
yield as result a number which has the same property as the departing
number (being a natural number) but is not natural number... Also
induction and inference cannot work in such a context.
For you to be able to use the word all, you must define the domain
of that word. If you do not define the domain, then it will be
impossible for me and all other humans to understand what you are
talking about.
Well you are the first and only human I know who don't understand
all as everybody else does.
Quentin Anciaux
--
Torgny Tholerus
--
All those moments will be lost in time, like tears in rain.
--
All those moments will be lost in time, like tears in rain.
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Re: The seven step-Mathematical preliminaries
Quentin Anciaux wrote: You have to explain why the exception is needed in the first place... The rule is true until the rule is not true anymore, ok but you have to explain for what sufficiently large N the successor function would yield next 0 and why or to add that N and that exception to the successor function as axiom, if not you can't avoid N+1. But torgny doesn't evacuate N+1, merely it allows his set to grows undefinitelly as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense , is a natural number but not part of the set of natural number, this is non-sense, assuming your special successor rule BIGGEST+1 simply does not exists at all. I can understand this overflow successor function for a finite data type or a real machine registe but not for N. The successor function is simple, if you want it to have an exception at biggest you should justify it. You don't justify definitions. How would you justify Peano's axioms as being the right ones? You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. Torgny is denying that and pointing out that we cannot know of infinite sets that exist independent of their definition because we cannot extensively define an infinite set, we can only know about it what we can prove from its definition. So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical objects. The first however is more definite than the second, since Godel's theorems don't apply. Which one is called the *natural* numbers is a convention which might not have any practical consequences for sufficiently large BIGGEST. Brent --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Tue, 9 Jun 2009 18:38:23 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. What's a value set? And why do you say we must associate it in this way? Do you have a philosophical argument for this must, or is it just an edict that reflects your personal aesthetic preferences? Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. *I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? When I talk about universe here, I do not mean our physical universe. What I mean is something that can be called everything. It includes all objects in our physical universe, as well as all symbols and all words and all numbers and all sets and all other universes. It includes everything you can use the word all about. For you to be able to use the word all, you must define the domain of that word. If you do not define the domain, then it will be impossible for me and all other humans to understand what you are talking about. OK, so how do you say I should define this type of universe? Unless you are demanding that I actually give you a list which spells out every symbol-string that qualifies as a member, can't I simply provide an abstract *rule* that would allow someone to determine in principle if a particular symbol-string they are given qualifies? Or do you have a third alternative besides spelling out every member or giving an abstract rule? Jesse --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/9 Brent Meeker meeke...@dslextreme.com: Quentin Anciaux wrote: You have to explain why the exception is needed in the first place... The rule is true until the rule is not true anymore, ok but you have to explain for what sufficiently large N the successor function would yield next 0 and why or to add that N and that exception to the successor function as axiom, if not you can't avoid N+1. But torgny doesn't evacuate N+1, merely it allows his set to grows undefinitelly as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense , is a natural number but not part of the set of natural number, this is non-sense, assuming your special successor rule BIGGEST+1 simply does not exists at all. I can understand this overflow successor function for a finite data type or a real machine registe but not for N. The successor function is simple, if you want it to have an exception at biggest you should justify it. You don't justify definitions. then you say it is an axiom, no problem with that. How would you justify Peano's axioms as being the right ones? You don't, and either I misexpressed myself or you did not understood. You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. No, I have a definition for a set called the set of natural number, this set is defined by the peano's axioms and the set defined by these axioms is unbounded and it is called the set of natural number. Any upper limit bounded set containing natural number is not N but a subset of N. http://en.wikipedia.org/wiki/Natural_number#Formal_definitions The set Torgny is talking about is not N, like a dog is not a cat, he can call it whatever he likes but not N. But merely what I want to point out is that the definition he use is inconsistent unlike yours which is simply modulo arithmetics. http://en.wikipedia.org/wiki/Modular_arithmetic Torgny is denying that and pointing out that we cannot know of infinite sets that exist independent of their definition because we cannot extensively define an infinite set, we can only know about it what we can prove from its definition. So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical objects. The first however is more definite than the second, since Godel's theorems don't apply. Which one is called the *natural* numbers is a convention which might not have any practical consequences for sufficiently large BIGGEST. Brent -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/9 Quentin Anciaux allco...@gmail.com: 2009/6/9 Brent Meeker meeke...@dslextreme.com: Quentin Anciaux wrote: You have to explain why the exception is needed in the first place... The rule is true until the rule is not true anymore, ok but you have to explain for what sufficiently large N the successor function would yield next 0 and why or to add that N and that exception to the successor function as axiom, if not you can't avoid N+1. But torgny doesn't evacuate N+1, merely it allows his set to grows undefinitelly as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense , is a natural number but not part of the set of natural number, this is non-sense, assuming your special successor rule BIGGEST+1 simply does not exists at all. I can understand this overflow successor function for a finite data type or a real machine registe but not for N. The successor function is simple, if you want it to have an exception at biggest you should justify it. You don't justify definitions. then you say it is an axiom, no problem with that. And your axiom can't just say there is a BIGGEST number without having a rule to either find it or discriminate it or setting the value arbitrarily. BIGGEST must be a well defined number not a boundary that you can't reach... because if it was the case you're no more an ultrafinitist and N is not a problem. How would you justify Peano's axioms as being the right ones? You don't, and either I misexpressed myself or you did not understood. You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. No, I have a definition for a set called the set of natural number, this set is defined by the peano's axioms and the set defined by these axioms is unbounded and it is called the set of natural number. Any upper limit bounded set containing natural number is not N but a subset of N. http://en.wikipedia.org/wiki/Natural_number#Formal_definitions The set Torgny is talking about is not N, like a dog is not a cat, he can call it whatever he likes but not N. But merely what I want to point out is that the definition he use is inconsistent unlike yours which is simply modulo arithmetics. http://en.wikipedia.org/wiki/Modular_arithmetic Torgny is denying that and pointing out that we cannot know of infinite sets that exist independent of their definition because we cannot extensively define an infinite set, we can only know about it what we can prove from its definition. So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical objects. The first however is more definite than the second, since Godel's theorems don't apply. Which one is called the *natural* numbers is a convention which might not have any practical consequences for sufficiently large BIGGEST. Brent -- All those moments will be lost in time, like tears in rain. -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
A good model of the naturalist math that Torgny is talking about is the overflow mechanism in computers. For example in a 64 bit machine you may define overflow for positive integers as 2^^64 -1. If negative integers are included then the biggest positive could be 2^^32-1. Torgny would also have to define the operations +, - x / with specific exceptions for overflow. The concept of BIGGEST needs to be tied with _the kind of operations you want to apply to_ the numbers. George Brent Meeker wrote: Quentin Anciaux wrote: You have to explain why the exception is needed in the first place... The rule is true until the rule is not true anymore, ok but you have to explain for what sufficiently large N the successor function would yield next 0 and why or to add that N and that exception to the successor function as axiom, if not you can't avoid N+1. But torgny doesn't evacuate N+1, merely it allows his set to grows undefinitelly as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense , is a natural number but not part of the set of natural number, this is non-sense, assuming your special successor rule BIGGEST+1 simply does not exists at all. I can understand this overflow successor function for a finite data type or a real machine registe but not for N. The successor function is simple, if you want it to have an exception at biggest you should justify it. You don't justify definitions. How would you justify Peano's axioms as being the right ones? You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. Torgny is denying that and pointing out that we cannot know of infinite sets that exist independent of their definition because we cannot extensively define an infinite set, we can only know about it what we can prove from its definition. So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical objects. The first however is more definite than the second, since Godel's theorems don't apply. Which one is called the *natural* numbers is a convention which might not have any practical consequences for sufficiently large BIGGEST. Brent --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries You don't justify definitions. How would you justify Peano's axioms as being the right ones? You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. What do you mean by exists in this context? What would it mean to have a well-defined, non-contradictory definition of some mathematical objects, and yet for those mathematical objects not to exist? --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries You don't justify definitions. How would you justify Peano's axioms as being the right ones? You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. What do you mean by exists in this context? What would it mean to have a well-defined, non-contradictory definition of some mathematical objects, and yet for those mathematical objects not to exist? A good question. But if one talks about some mathematical object, like the natural numbers, having properties that are unprovable from their defining set of axioms then it seems that one has assumed some kind of existence apart from the particular definition. Everybody believes arithmetic, per Peano's axioms, is consistent, but we know that can't be proved from Peano's axioms. So it seems we are assigning (or betting on, as Bruno might say) more existence than is implied by the definition. When Quentin insists that Peano's axioms are the right ones for the natural numbers, he is either just making a statement about language conventions, or he has an idea of the natural numbers that is independent of the axioms and is saying the axioms pick out the right set of natural numbers. Brent --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/10 Brent Meeker meeke...@dslextreme.com: Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries You don't justify definitions. How would you justify Peano's axioms as being the right ones? You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. What do you mean by exists in this context? What would it mean to have a well-defined, non-contradictory definition of some mathematical objects, and yet for those mathematical objects not to exist? A good question. But if one talks about some mathematical object, like the natural numbers, having properties that are unprovable from their defining set of axioms then it seems that one has assumed some kind of existence apart from the particular definition. Everybody believes arithmetic, per Peano's axioms, is consistent, but we know that can't be proved from Peano's axioms. So it seems we are assigning (or betting on, as Bruno might say) more existence than is implied by the definition. When Quentin insists that Peano's axioms are the right ones for the natural numbers, he is either just making a statement about language conventions, or he has an idea of the natural numbers that is independent of the axioms and is saying the axioms pick out the right set of natural numbers. Brent No I'm actually saying that peano's axiom define the abstract rules which permits to know if a number is a natural number or not. A number is a natural number if it satisfies peano's axiom... so by definition the set created by the numbers satisfying these rules is the set of all natural numbers. So if you change the rules, you change the set hence the new set(s) created by your new rules (axiom) is(are) not the same set(s) than the one denoted by peano's axioms hence it is not N and can't be by definition. The mathematical object you define with your new rules is not the same. And please note that modulo arithmetic is not the problem here. Torgny is not talking about that, he said BIGGEST+1 is not in the set N, but BIGGEST+1 is a natural number (Question1: What is a natural number ?, Question2: How can a natural number not be in the set of **all** natural numbers ?). With your version with modulo(BIGGEST), BIGGEST+1 is in the previously defined set, it is '0'. And in your version BIGGEST+1 doesn't satisfy that it is strictly bigger than BIGGEST, but in Torgny version it does. Regards, Quentin -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Tue, 9 Jun 2009 15:22:10 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries You don't justify definitions. How would you justify Peano's axioms as being the right ones? You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. What do you mean by exists in this context? What would it mean to have a well-defined, non-contradictory definition of some mathematical objects, and yet for those mathematical objects not to exist? A good question. But if one talks about some mathematical object, like the natural numbers, having properties that are unprovable from their defining set of axioms then it seems that one has assumed some kind of existence apart from the particular definition. Isn't this based on the idea that there should be an objective truth about every well-formed proposition about the natural numbers even if the Peano axioms cannot decide the truth about all propositions? I think that the statements that cannot be proved are disproved would all be ones of the type for all numbers with property X, Y is true or there exists a number (or some finite group of numbers) with property X (i.e. propositions using either the 'for all' or 'there exists' universal quantifiers in logic, with variables representing specific numbers or groups of numbers). So to believe these statements are objectively true basically means there would be a unique way to extend our judgment of the truth-values of propositions from the judgments already given by the Peano axioms, in such a way that if we could flip through all the infinite propositions judged true by the Peano axioms, we would *not* find an example of a proposition like for this specific number N with property X, Y is false (which would disprove the 'for all' proposition above), and likewise we would not find that for every possible number (or group of numbers) N, the Peano axioms proved a proposition like number N does not have property X (which would disprove the 'there exists' proposition above). We can't actual flip through an infinite number of propositions in a finite time of course, but if we had a hypercomputer that could do so (which is equivalent to the notion of a hypercomputer that can decide in finite time if any given Turing program halts or not), then I think we'd have a well-defined notion of how to program it to decide the truth of every for all or there exists proposition in a way that's compatible with the propositions already proved by the Peano axioms. If I'm right about that, it would lead naturally to the idea of something like a unique consistent extension of the Peano axioms (not a real technical term, I just made up this phrase, but unless there's an error in my reasoning I imagine mathematicians have some analogous notion...maybe Bruno knows?) which assigns truth values to all the well-formed propositions that are undecidable by the Peano axioms themselves. So this would be a natural way of understanding the idea of truths about the natural numbers that are not decidable by the Peano axioms. Of course even if the notion of a unique consistent extension of certain types of axiomatic systems is well-defined, it would only make sense for axiomatic systems that are consistent in the first place. I guess in judging the question of the consistency of the Peano axioms, we must rely on some sort of ill-defined notion of our understanding of how the axioms should represent true statements about things like counting discrete objects. For example, we understand that the order you count a group of discrete objects doesn't affect the total number, which is a convincing argument for believing that A + B = B + A regardless of what numbers you choose for A and B. Likewise, we understand that multiplying A * B can be thought of in terms of a square array of discrete objects with the horizontal side having A objects and the vertical side having B objects, and we can see that just by rotating this you get a square array with B on the horizontal side and A on the vertical side, so if we believe that just mentally rotating an array of discrete objects won't change the number in the array that's a good argument for believing A * B = B * A. So thinking along these lines, as long as we don't believe that true statements about counting collections of discrete objects could ever lead to logical contradictions, we should believe the same for the Peano axioms. Jesse --~--~-~--~~~---~--~~ You received this message because you are subscribed
Re: The seven step-Mathematical preliminaries
Jesse Mazer wrote: Date: Tue, 9 Jun 2009 15:22:10 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries You don't justify definitions. How would you justify Peano's axioms as being the right ones? You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. What do you mean by exists in this context? What would it mean to have a well-defined, non-contradictory definition of some mathematical objects, and yet for those mathematical objects not to exist? A good question. But if one talks about some mathematical object, like the natural numbers, having properties that are unprovable from their defining set of axioms then it seems that one has assumed some kind of existence apart from the particular definition. Isn't this based on the idea that there should be an objective truth about every well-formed proposition about the natural numbers even if the Peano axioms cannot decide the truth about all propositions? I think that the statements that cannot be proved are disproved would all be ones of the type for all numbers with property X, Y is true or there exists a number (or some finite group of numbers) with property X (i.e. propositions using either the 'for all' or 'there exists' universal quantifiers in logic, with variables representing specific numbers or groups of numbers). So to believe these statements are objectively true basically means there would be a unique way to extend our judgment of the truth-values of propositions from the judgments already given by the Peano axioms, in such a way that if we could flip through all the infinite propositions judged true by the Peano axioms, we would *not* find an example of a proposition like for this specific number N with property X, Y is false (which would disprove the 'for all' proposition above), and likewise we would not find that for every possible number (or group of numbers) N, the Peano axioms proved a proposition like number N does not have property X (which would disprove the 'there exists' proposition above). We can't actual flip through an infinite number of propositions in a finite time of course, but if we had a hypercomputer that could do so (which is equivalent to the notion of a hypercomputer that can decide in finite time if any given Turing program halts or not), then I think we'd have a well-defined notion of how to program it to decide the truth of every for all or there exists proposition in a way that's compatible with the propositions already proved by the Peano axioms. If I'm right about that, it would lead naturally to the idea of something like a unique consistent extension of the Peano axioms (not a real technical term, I just made up this phrase, but unless there's an error in my reasoning I imagine mathematicians have some analogous notion...maybe Bruno knows?) which assigns truth values to all the well-formed propositions that are undecidable by the Peano axioms themselves. So this would be a natural way of understanding the idea of truths about the natural numbers that are not decidable by the Peano axioms. I think Godel's imcompleteness theorem already implies that there must be non-unique extensions, (e.g. maybe you can add an axiom either that there are infinitely many pairs of primes differing by two or the negative of that). That would seem to be a reductio against the existence of a hypercomputer that could decide these propositions by inspection. Of course even if the notion of a unique consistent extension of certain types of axiomatic systems is well-defined, it would only make sense for axiomatic systems that are consistent in the first place. I guess in judging the question of the consistency of the Peano axioms, we must rely on some sort of ill-defined notion of our understanding of how the axioms should represent true statements about things like counting discrete objects. For example, we understand that the order you count a group of discrete objects doesn't affect the total number, which is a convincing argument for believing that A + B = B + A regardless of what numbers you choose for A and B. Likewise, we understand that multiplying A * B can be thought of in terms of a square array of discrete objects with the horizontal side having A objects and the vertical side having B objects, and we can see that just by rotating this you get a square array with B on the horizontal side and A on the vertical side, so if we believe that just mentally rotating an array
RE: The seven step-Mathematical preliminaries
Date: Tue, 9 Jun 2009 17:20:39 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer wrote: Date: Tue, 9 Jun 2009 15:22:10 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries You don't justify definitions. How would you justify Peano's axioms as being the right ones? You are just confirming my point that you are begging the question by assuming there is a set called the natural numbers that exists independently of it's definition and it satisfies Peano's axioms. What do you mean by exists in this context? What would it mean to have a well-defined, non-contradictory definition of some mathematical objects, and yet for those mathematical objects not to exist? A good question. But if one talks about some mathematical object, like the natural numbers, having properties that are unprovable from their defining set of axioms then it seems that one has assumed some kind of existence apart from the particular definition. Isn't this based on the idea that there should be an objective truth about every well-formed proposition about the natural numbers even if the Peano axioms cannot decide the truth about all propositions? I think that the statements that cannot be proved are disproved would all be ones of the type for all numbers with property X, Y is true or there exists a number (or some finite group of numbers) with property X (i.e. propositions using either the 'for all' or 'there exists' universal quantifiers in logic, with variables representing specific numbers or groups of numbers). So to believe these statements are objectively true basically means there would be a unique way to extend our judgment of the truth-values of propositions from the judgments already given by the Peano axioms, in such a way that if we could flip through all the infinite propositions judged true by the Peano axioms, we would *not* find an example of a proposition like for this specific number N with property X, Y is false (which would disprove the 'for all' proposition above), and likewise we would not find that for every possible number (or group of numbers) N, the Peano axioms proved a proposition like number N does not have property X (which would disprove the 'there exists' proposition above). We can't actual flip through an infinite number of propositions in a finite time of course, but if we had a hypercomputer that could do so (which is equivalent to the notion of a hypercomputer that can decide in finite time if any given Turing program halts or not), then I think we'd have a well-defined notion of how to program it to decide the truth of every for all or there exists proposition in a way that's compatible with the propositions already proved by the Peano axioms. If I'm right about that, it would lead naturally to the idea of something like a unique consistent extension of the Peano axioms (not a real technical term, I just made up this phrase, but unless there's an error in my reasoning I imagine mathematicians have some analogous notion...maybe Bruno knows?) which assigns truth values to all the well-formed propositions that are undecidable by the Peano axioms themselves. So this would be a natural way of understanding the idea of truths about the natural numbers that are not decidable by the Peano axioms. I think Godel's imcompleteness theorem already implies that there must be non-unique extensions, (e.g. maybe you can add an axiom either that there are infinitely many pairs of primes differing by two or the negative of that). That would seem to be a reductio against the existence of a hypercomputer that could decide these propositions by inspection. I think I remember reading in one of Roger Penrose's books that there is a difference between an ordinary consistency condition (which just means that no two propositions explicitly contradict each other) and omega-consistency--see http://en.wikipedia.org/wiki/Omega-consistent_theory . I can't quite follow the details, but I'm guessing the condition means (or at least includes) something like the idea that if you have a statement of the form there exists a number (or set of numbers) with property X then there must actually be some other proposition describing a particular number (or set of numbers) does in fact have this property. The fact that you can add either a Godel statement or its negation to the Peano axioms without creating a contradiction (as long as the Peano axioms are not inconsistent) may not mean you can add either one and still have an omega-consistent theory; if that's true, would
Re: The seven step-Mathematical preliminaries
On Sat, Jun 06, 2009 at 10:22:11AM -0700, Brent Meeker wrote:
I wonder if anyone has tried work with a theory of finite numbers: where
BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers?
Brent
The numbers {0,...,p-1} with p prime, and addition and multiplication
given modulo p (ie
a plus b = (a+b) mod p
a times b = (ab) mod p
)
is an interesting mathematical object known as a finite field (or
Galois field) -
http://en.wikipedia.org/wiki/Finite_field
Interesting examples of infinite fields are those quite familiar to
you: rational, real and complex numbers.
It might make sense for Torgny to work with a Galois field for some
large but unnamed prime :)
Cheers
--
Prof Russell Standish Phone 0425 253119 (mobile)
Mathematics
UNSW SYDNEY 2052 hpco...@hpcoders.com.au
Australiahttp://www.hpcoders.com.au
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Re: The seven step-Mathematical preliminaries 2
Marty,
On 07 Jun 2009, at 02:03, Brent Meeker wrote:
m.a. wrote:
*Okay, so is it true to say that things written in EXTENSION are
never
in formula style but are translated into formulas when we put them
into INTENSION form? You can see that my difficulty with math
arises from an inability to master even the simplest definitions.
marty a.*
It's not that technical. I could define the set of books on my
shelf by
giving a list of titles: The Comprehensible Cosmos, Set Theory and
It's Philosophy, Overshoot, Quintessence. That would be a
definition by extension. Or I could point to them in succession and
say, That and that and that and that. which would be a definition by
ostension. Or I could just say, The books on my shelf. which is a
definition by intension. An intensional definition is a descriptive
phrase with an implicit variable, which in logic you might write as:
The
set of things x such that x is a book and x is on my shelf.
This is a good point. A set is just a collection of objects seen as a
whole.
A definition in extension of a set is just a listing, finite or
infinite, of its elements.
Like in A = {1, 3, 5}, or B = {2, 4, 6, 8, 10, ...}.
A definition in intension of a set consists in giving the typical
defining property of the elements of the set.
Like in C= the set of odd numbers which are smaller than 6. Or D =
the set of even numbers.
In this case you see that A is the same set as C? And B is the same
set as D.
Now in mathematics we often use abbreviation. So, for example, instead
of saying: the set of even numbers, we will write
{x such-that x is even}.
OK?
Bruno
Suppose,
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
Thank you, Brent, This is quite clear. Hopefully I can apply it as clearly to Bruno's examples.marty a. - Original Message - From: Brent Meeker meeke...@dslextreme.com To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 8:03 PM Subject: Re: The seven step-Mathematical preliminaries 2 m.a. wrote: *Okay, so is it true to say that things written in EXTENSION are never in formula style but are translated into formulas when we put them into INTENSION form? You can see that my difficulty with math arises from an inability to master even the simplest definitions. marty a.* It's not that technical. I could define the set of books on my shelf by giving a list of titles: The Comprehensible Cosmos, Set Theory and It's Philosophy, Overshoot, Quintessence. That would be a definition by extension. Or I could point to them in succession and say, That and that and that and that. which would be a definition by ostension. Or I could just say, The books on my shelf. which is a definition by intension. An intensional definition is a descriptive phrase with an implicit variable, which in logic you might write as: The set of things x such that x is a book and x is on my shelf. Brent --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Bruno,
Yes, this seems very clear and will be helpful to refer back to if
necessary. m.a.
- Original Message -
From: Bruno Marchal marc...@ulb.ac.be
To: everything-list@googlegroups.com
Sent: Sunday, June 07, 2009 4:33 AM
Subject: Re: The seven step-Mathematical preliminaries 2
Marty,
On 07 Jun 2009, at 02:03, Brent Meeker wrote:
m.a. wrote:
*Okay, so is it true to say that things written in EXTENSION are
never
in formula style but are translated into formulas when we put them
into INTENSION form? You can see that my difficulty with math
arises from an inability to master even the simplest definitions.
marty a.*
It's not that technical. I could define the set of books on my
shelf by
giving a list of titles: The Comprehensible Cosmos, Set Theory and
It's Philosophy, Overshoot, Quintessence. That would be a
definition by extension. Or I could point to them in succession and
say, That and that and that and that. which would be a definition by
ostension. Or I could just say, The books on my shelf. which is a
definition by intension. An intensional definition is a descriptive
phrase with an implicit variable, which in logic you might write as:
The
set of things x such that x is a book and x is on my shelf.
This is a good point. A set is just a collection of objects seen as a
whole.
A definition in extension of a set is just a listing, finite or
infinite, of its elements.
Like in A = {1, 3, 5}, or B = {2, 4, 6, 8, 10, ...}.
A definition in intension of a set consists in giving the typical
defining property of the elements of the set.
Like in C= the set of odd numbers which are smaller than 6. Or D =
the set of even numbers.
In this case you see that A is the same set as C? And B is the same
set as D.
Now in mathematics we often use abbreviation. So, for example, instead
of saying: the set of even numbers, we will write
{x such-that x is even}.
OK?
Bruno
Suppose,
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
On Sat, Jun 6, 2009 at 4:20 PM, Jesse Mazer laserma...@hotmail.com wrote: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: [[[ Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed.]]] Why humans, specifically? What if an alien could conceptualize a larger number? For that matter, since you deny any special role to consciousness, why should it have anything to do with the conceptualizations of beings with brains? A volume of space isn't normally said to conceptualize the number of atoms contained in that volume, but why should that number be any less real than the largest number that's been conceptualized by a biological brain? *JohnM:* *Jesse, * *you don't have to go out to 'aliens', just eliminate the format possible as of 2009. Our un-alien species is well capable of learning (compare to 2000BC) and whatever is restricted today as 'impossible' may be everyday's bread after tomorrow. You are absolutely right - even as of today. * *Especially in your next reply-par below.* Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. You can only create a finite number of proposition with finite length during your lifetime. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. But you said earlier that as long as we admit only a finite collection of numbers, we can prove the consistency of mathematics involving only those numbers. Well, how can we prove that? If we only show that all the propositions we have generated to date are consistent, how do we know the next proposition we generate won't involve an inconsistency? Presumably you are implicitly suggesting there should be some upper limit on the number of propositions about the numbers as well as on the numbers themselves, but if you define this limit in terms of how many a human could generate in their lifetime, we get back to problems like what if some other being (genetically engineered humans, say) would have a longer lifetime, or what if we built a computer that generated propositions much faster than a human could and checked their consistency automatically, etc. or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. What's a value set? And why do you say we must associate it in this way? Do you have a philosophical argument for this must, or is it just an edict that reflects your personal aesthetic preferences? Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. *I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? One of the first thing kids learn about number is that if you count some collection of objects, it doesn't matter what order you count them in, the final number you get will be the same regardless of the order (i.e. it doesn't matter which you point to when you say 1 and which you point to when you say 2, as long as you point to each object exactly once). Also, am I understanding correctly in thinking you don't believe there can
Re: The seven step-Mathematical preliminaries 2
*Bruno et. al., Good news! I have discovered that the math symbols copy faithfully here in my Thunderbird email.* *Henceforth, I will open all list letters here. Please refresh my memory for the following symbols:* * 1. The ***?** *is called_and means__ 2. The***?** *is called___*_*and means__ 3. The ***? is called__and means ** -* Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ? ? A = ? ? B = A ? ? = B ? ? = N ? ? = B ? ? = ? ? B = ? ? ? = ? ? ? = * --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Bravo Thunderbird!
On 07 Jun 2009, at 18:39, m.a. wrote:
Bruno et. al.,
Good news! I have discovered that the math
symbols copy faithfully here in my Thunderbird email. Henceforth, I
will open all list letters here. Please refresh my memory for the
following symbols:
1. The ∅ is called___THE EMPTY SET_and means__THE SET
WITH NO ELEMENTS
The empty set described in extension: { }
The empty set described in intension. Well, let me think. The set of
french which are bigger than 42 km tall.
A cynical definition would be: the set of honest politicians.
A mathematical one: the set of x such that x is different from x.
It is just the set which has no elements. It is empty.
2. The∪ is calledUNION__and means: A ∪ B__= {x
such-that x belongs to A or x belongs to B};
A u B is the set obtained by doing the union of A and B.
3. The ∩ is called_INTERSECTIONand means__A ∩ B__=
{x such-that x belongs to A andr x belongs to B}; A u B is the
set obtained by doing the intersection of A and B. It is the set of
elements which are in both A and B._
Examples:
{1, 2, 3} ∩ {2, 4, 3} = {2, 3}
{1, 2, 3} u {2, 4, 3} = {1, 2, 3, 4}
{1, 2, 3} ∩ {4, 5, 6} = ∅
{1, 2, 3} u {4, 5, 6} = {1, 2, 3, 4, 5, 6}
OK?
Bruno
- Original Message -
From: Bruno Marchal
To: everything-list@googlegroups.com
Sent: Wednesday, June 03, 2009 1:15 PM
Subject: Re: The seven step-Mathematical preliminaries 2
∅ ∪ A =
∅ ∪ B =
A ∪ ∅ =
B ∪ ∅ =
N ∩ ∅ =
B ∩ ∅ =
∅ ∩ B =
∅ ∩ ∅ =
∅ ∪ ∅ =
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
Marty, Kim,
I realize that, now, the message I have just sent does not have the
right symbols. Apparently my computer does not understand the
Thunderbird!
From now on I will use capital words for the mathematical symbols.
And I will write mathematical expression in bold.
For examples:
{1, 2, 3} INTERSECTION {2, 4, 3} = {2, 3}
{1, 2, 3} UNION {2, 4, 3} = {1, 2, 3, 4}
{1, 2, 3} INTERSECTION {4, 5, 6} = EMPTY
{1, 2, 3} UNION {4, 5, 6} = {1, 2, 3, 4, 5, 6}
All right? Mathematics will get a FORTRAN look but this is not
important, OK? It is just the look. I will do a summary of what we
have seen so far.
With those notions you should be able to invent exercises by yourself.
Invent simple sets and compute their union, and intersection.
Remenber that the goal consists in building a mathematical shortcut
toward a thorugh understanding of step seven. In particular the goal
will be to get an idea of a computation is, and what is the difference
between a mathemarical computation and a mathematical description of a
computation. It helps for the step 8 too.
Marty, have a nice holiday,
Kim, ah ah ... we have two weeks to digest what has been said so far
(which is not enormous), OK?
Bruno
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev:
Date: Fri, 5 Jun 2009 08:33:47 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0,
1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers.
Here you're just contradicting yourself. If you say BIGGEST+1 is then
a natural number, that just proves that the set N was not in fact the
set of all natural numbers. The alternative would be to say
BIGGEST+1 is *not* a natural number, but then you need to provide a
definition of natural number that would explain why this is the case.
It depends upon how you define natural number. If you define it by: n
is a natural number if and only if n belongs to N, the set of all
natural numbers, then of course BIGGEST+1 is *not* a natural number. In
that case you have to call BIGGEST+1 something else, maybe unnatural
number.
--
Torgny Tholerus
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RE: The seven step-Mathematical preliminaries
Date: Sat, 6 Jun 2009 16:48:21 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev:
Date: Fri, 5 Jun 2009 08:33:47 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0,
1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers.
Here you're just contradicting yourself. If you say BIGGEST+1 is then
a natural number, that just proves that the set N was not in fact the
set of all natural numbers. The alternative would be to say
BIGGEST+1 is *not* a natural number, but then you need to provide a
definition of natural number that would explain why this is the case.
It depends upon how you define natural number. If you define it by: n
is a natural number if and only if n belongs to N, the set of all
natural numbers, then of course BIGGEST+1 is *not* a natural number. In
that case you have to call BIGGEST+1 something else, maybe unnatural
number.
OK, but then you need to define what you mean by N, the set of all natural
numbers. Specifically you need to say what number is BIGGEST. Is it
arbitrary? Can I set BIGGEST = 3, for example? Or do you have some
philosophical ideas related to what BIGGEST is, like the number of particles in
the universe or the largest number any human can conceptualize?
Also, any comment on my point about there being an infinite number of possible
propositions about even a finite set, or about my question about whether you
have any philosophical/logical argument for saying all sets must be finite, as
opposed to it just being a sort of aesthetic preference on your part? Do you
think there is anything illogical or incoherent about defining a set in terms
of a rule that takes any input and decides whether it's a member of the set or
not, such that there may be no upper limit on the number of possible inputs
that the rule would define as being members? (such as would be the case for the
rule 'n is a natural number if n=1 or if n is equal to some other natural
number+1')
Jesse
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Re: The seven step-Mathematical preliminaries
Torgny Tholerus wrote:
Jesse Mazer skrev:
Date: Fri, 5 Jun 2009 08:33:47 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0,
1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers.
Here you're just contradicting yourself. If you say BIGGEST+1 is then
a natural number, that just proves that the set N was not in fact the
set of all natural numbers. The alternative would be to say
BIGGEST+1 is *not* a natural number, but then you need to provide a
definition of natural number that would explain why this is the case.
It depends upon how you define natural number. If you define it by: n
is a natural number if and only if n belongs to N, the set of all
natural numbers, then of course BIGGEST+1 is *not* a natural number. In
that case you have to call BIGGEST+1 something else, maybe unnatural
number.
I wonder if anyone has tried work with a theory of finite numbers: where
BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers?
Brent
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Re: The seven step-Mathematical preliminaries
I wonder if anyone has tried work with a theory of finite numbers: where BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers? There is a group of faculty who address this problem directly in my department. But any general-purpose computer can emulate true, unlimited natural numbers (which is what people often do, rather than relying on bounded ints). The only real limitations that make computer not-equal-to Turing machine are memory and the limited patience of humans. This is one reason why people spend more time researching P vs. NP than artificially-imposed limits. When you add bounds to numbers it requires additional proof obligations, which makes it more difficult to prove things. And you can't directly prove anything about numbers that exist outside the bounds under which you're working. Anna --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~--- --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. You can only create a finite number of proposition with finite length during your lifetime. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. But as soon you go outside universe, you must be carefull with what substitutions you do. If you have all quantified with all object inside the universe, you can not substitute it with an object outside the universe, because that object was not included in the original statement. as opposed to it just being a sort of aesthetic preference on your part? Do you think there is anything illogical or incoherent about defining a set in terms of a rule that takes any input and decides whether it's a member of the set or not, such that there may be no upper limit on the number of possible inputs that the rule would define as being members? (such as would be the case for the rule 'n is a natural number if n=1 or if n is equal to some other natural number+1') In the last sentence you have an implicite all: The full sentence would be: For all n in the universe hold that n is a natural number if n=1 or if n is equal to some other natural number+1. And you may now be able to understand, that if the number of objects in the universe is finite, then this sentence will just define a finite set. -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. Why humans, specifically? What if an alien could conceptualize a larger number? For that matter, since you deny any special role to consciousness, why should it have anything to do with the conceptualizations of beings with brains? A volume of space isn't normally said to conceptualize the number of atoms contained in that volume, but why should that number be any less real than the largest number that's been conceptualized by a biological brain? Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. You can only create a finite number of proposition with finite length during your lifetime. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. But you said earlier that as long as we admit only a finite collection of numbers, we can prove the consistency of mathematics involving only those numbers. Well, how can we prove that? If we only show that all the propositions we have generated to date are consistent, how do we know the next proposition we generate won't involve an inconsistency? Presumably you are implicitly suggesting there should be some upper limit on the number of propositions about the numbers as well as on the numbers themselves, but if you define this limit in terms of how many a human could generate in their lifetime, we get back to problems like what if some other being (genetically engineered humans, say) would have a longer lifetime, or what if we built a computer that generated propositions much faster than a human could and checked their consistency automatically, etc. or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. What's a value set? And why do you say we must associate it in this way? Do you have a philosophical argument for this must, or is it just an edict that reflects your personal aesthetic preferences? Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. *I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? One of the first thing kids learn about number is that if you count some collection of objects, it doesn't matter what order you count them in, the final number you get will be the same regardless of the order (i.e. it doesn't matter which you point to when you say 1 and which you point to when you say 2, as long as you point to each object exactly once). Also, am I understanding correctly in thinking you don't believe there can be truths about numbers independent of what humans actually know about them (i.e. there is no truth about the sum of two very large numbers unless some human has actually calculated that sum at one point)? If in fact you don't believe there are truths about numbers independent of human thoughts about them, why do you think there can be truths about the physical universe which humans don't know about? For example, is there a truth about the surface topography of some planet
Re: The seven step-Mathematical preliminaries 2
Marty, Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~--- http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Jesse Mazer wrote: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. Why humans, specifically? What if an alien could conceptualize a larger number? For that matter, since you deny any special role to consciousness, why should it have anything to do with the conceptualizations of beings with brains? A volume of space isn't normally said to conceptualize the number of atoms contained in that volume, but why should that number be any less real than the largest number that's been conceptualized by a biological brain? Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. You can only create a finite number of proposition with finite length during your lifetime. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. But you said earlier that as long as we admit only a finite collection of numbers, we can prove the consistency of mathematics involving only those numbers. Well, how can we prove that? If we only show that all the propositions we have generated to date are consistent, how do we know the next proposition we generate won't involve an inconsistency? Presumably you are implicitly suggesting there should be some upper limit on the number of propositions about the numbers as well as on the numbers themselves, but if you define this limit in terms of how many a human could generate in their lifetime, we get back to problems like what if some other being (genetically engineered humans, say) would have a longer lifetime, or what if we built a computer that generated propositions much faster than a human could and checked their consistency automatically, etc. or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. What's a value set? And why do you say we must associate it in this way? Do you have a philosophical argument for this must, or is it just an edict that reflects your personal aesthetic preferences? Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. *I* certainly don't define numbers in terms of any specific mapping between numbers and objects in the universe, it seems like a rather strange notion--shall we have arguments over whether the number 113485 should be associated with this specific shoelace or this specific kangaroo? One of the first thing kids learn about number is that if you count some collection of objects, it doesn't matter what order you count them in, the final number you get will be the same regardless of the order (i.e. it doesn't matter which you point to when you say 1 and which you point to when you say 2, as long as you point to each object exactly once). Also, am I understanding correctly in thinking you don't believe there can be truths about numbers independent of what humans actually know about them (i.e. there is no truth about the sum of two very large numbers unless some human has actually calculated that sum at one point)? If in fact you don't believe there are truths about numbers independent of human thoughts about them, why do you think there can be truths about the physical universe which
RE: The seven step-Mathematical preliminaries 2
If it helps, here's a screenshot of how the symbols are supposed to look: http://img34.imageshack.us/img34/3345/picture2uzk.png From: marc...@ulb.ac.be To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries 2 Date: Sat, 6 Jun 2009 22:36:01 +0200 Marty, Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message -From: Bruno MarchalTo: everything-l...@googlegroups.comsent: Wednesday, June 03, 2009 1:15 PMSubject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A =∅ ∪ B =A ∪ ∅ =B ∪ ∅ =N ∩ ∅ =B ∩ ∅ =∅ ∩ B =∅ ∩ ∅ =∅ ∪ ∅ = ---To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~--- http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
(I'll be here till Tuesday.) Evidently, the symbol you are using for such that is being shown on my screen as a small rectangle. In the copy below, I see two rectangles before the A=, two before the B=, two after the A, two after the B. The UNION symbol (inverted U) shows up but is followed by a rectangle in the next two examples and preceded by a rectangle in the last three. In checking a table of logic notaion, I find that the relation such that is designated by a reversed capital E. Is this the symbol you are using? m.a. - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 4:36 PM Subject: Re: The seven step-Mathematical preliminaries 2 We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~--- http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Bruno,
I've encountered some difficulty with the examples below. You say
that in extension describes exhaustion or quasi-exhaustion. And you give
the example: B = {3, 6, 9, 12, ... 99}.
Then you define in intension with exactly the same type of set:
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}.
Can you see the cause of my confusion? Incidentally, may I suggest
you use smaller than rather than more little than. Your English is
generally too good to include that kind of error. marty a.
- Original Message -
From: Bruno Marchal
To: everything-list@googlegroups.com
Sent: Wednesday, June 03, 2009 1:15 PM
Subject: Re: The seven step-Mathematical preliminaries 2
=== Intension and extension
In the case of finite and little set we have seen that we can define them
by exhaustion. This means we can give an explicit complete description of all
element of the set.
Example. A = {0, 1, 2, 77, 98, 5}
When the set is still finite and too big, or if we are lazy, we can sometimes
define the set by quasi exhaustion. This means we describe enough elements of
the set in a manner which, by requiring some good will and some imagination, we
can estimate having define the set.
Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case that we
meant the set of multiple of the number three, below 100.
A fortiori, when a set in not finite, that is, when the set is infinite, we
have to use either quasi-exhaustion, or we have to use some sentence or phrase
or proposition describing the elements of the set.
Definition.
I will say that a set is defined IN EXTENSIO, or simply, in extension, when
it is defined in exhaustion or quasi-exhaustion.
I will say that a set is defined IN INTENSIO, or simply in intension, with a
s, when it is defined by a sentence explaining the typical attribute of the
elements.
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily define A
in intension: A = the set of numbers which are even and more little than 100.
mathematician will condense this by the following:
A = {x such that x is even and little than 100} = {x ⎮ x is even x 100}.
⎮ is a special character, abbreviating such that, and I hope it goes
through the mail. If not I will use such that, or s.t., or things like that.
The expression {x ⎮ x is even} is literally read as: the set of object x,
(or number x if we are in a context where we talk about number) such that x is
even.
Exercise 1: Could you define in intension the following infinite set C =
{101, 103, 105, ...}
C = ?
Exercise 2: I will say that a natural number is a multiple of 4 if it can be
written as 4*y, for some y. For example 0 is a multiple of 4, (0 = 4*0), but
also 28, 400, 404, ... Could you define in extension the following set D = {x
⎮ x 10x is a multiple of 4}.
A last notational, but important symbol. Sets have elements. For example the
set A = {1, 2, 3} has three elements 1, 2 and 3. For saying that 3 is an
element of A in an a short way, we usually write 3 ∈ A. this is read as 3
belongs to A, or 3 is in A. Now 4 does not belong to A. To write this in a
short way, we will write 4 ∉ A, or we will write ¬ (4 ∈ A) or sometimes just
NOT(4 ∈ A). It is read: 4 does not belong to A, or: it is not the case that 4
belongs to A.
Having those notions and notations at our disposition we can speed up on the
notion of union and intersection.
The intersection of the sets A and B is the (new) set of those elements which
belongs to both A and B. Put in another way:
The intersection of the sets A with the set B is the set of those elements
which belongs to A and which belongs to B.
This new set, obtained from A and B is written A ∩ B, or A inter. B (in case
the special character doesn't go through).
With our notations we can write or define the intersection A ∩ B directly
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B, written A ∪
B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual logical or. p or q
is suppose to be true if p is true or q is true (or both are true). It is not
the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}.
Exercice 3.
Let N = {0, 1, 2, 3, ...}
Let A = {x ⎮ x 10}
Let B = {x ⎮ x is even}
Describe in extension (that is: exhaustion or quasi-exhaustion) the following
sets:
N ∪ A =
N ∪ B =
A ∪ B =
B ∪ A =
N ∩ A =
B ∩ A =
N ∩ B =
A ∩ B =
Exercice 4
Is it true that A ∩ B = B ∩ A, whatever A and B are?
Is it true that A ∪ B = B ∪ A, whatever A and B are?
Now, I could give you exercise so that you would be lead to discoveries, but
I prefer to be as simple and approachable as possible, and my goal is not even
to give you
Re: The seven step-Mathematical preliminaries
2009/6/6 Torgny Tholerus tor...@dsv.su.se: Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. What is the last number human can invent ? Your theory can't explain why addition works... If N is limited, then addition can and will (in human lifetime) create number which are still finite and not in N. N can be defined solelly as the successor function, you don't need anything else. You just have to assert that the function is true always. Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. Prove it please. You can only create a finite number of proposition with finite length during your lifetime. What is a lifetime . What is truth ? Either you CAN*** define a limit or you ***CAN'T***. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. EVERY*** ***MEMBER*** of the set ***N*** is FINITE* or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, My philosophical argument is about the mening of the word all. To be able to use that word, you must associate it with a value set. Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. But as soon you go outside universe, you must be carefull with what substitutions you do. If you have all quantified with all object inside the universe, you can not substitute it with an object outside the universe, because that object was not included in the original statement. as opposed to it just being a sort of aesthetic preference on your part? Do you think there is anything illogical or incoherent about defining a set in terms of a rule that takes any input and decides whether it's a member of the set or not, such that there may be no upper limit on the number of possible inputs that the rule would define as being members? (such as would be the case for the rule 'n is a natural number if n=1 or if n is equal to some other natural number+1') In the last sentence you have an implicite all: The full sentence would be: For all n in the universe hold that n is a natural number if n=1 or if n is equal to some other natural number+1. And you may now be able to understand, that if the number of objects in the universe is finite, then this sentence will just define a finite set. -- Torgny Tholerus I will read the rest (and others) email later unfortunatelly. -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
On 06 Jun 2009, at 23:54, m.a. wrote: (I'll be here till Tuesday.) Evidently, the symbol you are using for such that is being shown on my screen as a small rectangle. In the copy below, I see two rectangles before the A=, two before the B=, two after the A, two after the B. The UNION symbol (inverted U) shows up but is followed by a rectangle in the next two examples and preceded by a rectangle in the last three. In checking a table of logic notaion, I find that the relation such that is designated by a reversed capital E. Is this the symbol you are using? m.a. Yes, we have a problem. There should be no rectangles at all. We have to switch on english abbreviations. This explains the difficulty you did have with the union ... You could look on the archive, from here, http://www.mail-archive.com/everything-list@googlegroups.com/msg16531.html the symbols are correct on my computer, but we will think on easier mail symbols. Tell me if you see different symbols in the archive. Best, Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 4:36 PM Subject: Re: The seven step-Mathematical preliminaries 2 We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~--- http://iridia.ulb.ac.be/~marchal/ http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
I've encountered some difficulty with the examples below.
You say that in extension describes exhaustion or quasi-
exhaustion. And you give the example: B = {3, 6, 9, 12, ... 99}.
Then you define in intension with exactly the same type
of set: Example: Let A be the set {2, 4, 6, 8, 10, ... 100}.
I give A in extension there, but just to define it in intension after.
It is always the same set there. But I show its definition in
extension, to show the definition in intension after. You have to read
the to sentences.
Can you see the cause of my confusion?
It is always the same set. I give it in extension, and then in
intension.
Incidentally, may I suggest you use smaller than rather than
more little than. Your English is generally too good to include
that kind of error. marty a.
Well sure. Sometimes the correct expression just slip out from my
mind. smaller than is much better! Thanks for helping,
Bruno
- Original Message -
From: Bruno Marchal
To: everything-list@googlegroups.com
Sent: Wednesday, June 03, 2009 1:15 PM
Subject: Re: The seven step-Mathematical preliminaries 2
=== Intension and extension
In the case of finite and little set we have seen that we can
define them by exhaustion. This means we can give an explicit
complete description of all element of the set.
Example. A = {0, 1, 2, 77, 98, 5}
When the set is still finite and too big, or if we are lazy, we can
sometimes define the set by quasi exhaustion. This means we describe
enough elements of the set in a manner which, by requiring some good
will and some imagination, we can estimate having define the set.
Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case
that we meant the set of multiple of the number three, below 100.
A fortiori, when a set in not finite, that is, when the set is
infinite, we have to use either quasi-exhaustion, or we have to use
some sentence or phrase or proposition describing the elements of
the set.
Definition.
I will say that a set is defined IN EXTENSIO, or simply, in
extension, when it is defined in exhaustion or quasi-exhaustion.
I will say that a set is defined IN INTENSIO, or simply in
intension, with a s, when it is defined by a sentence explaining
the typical attribute of the elements.
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily
define A in intension: A = the set of numbers which are even and
more little than 100. mathematician will condense this by the
following:
A = {x such that x is even and little than 100} = {x ⎮ x is even
x 100}. ⎮ is a special character, abbreviating such that, and
I hope it goes through the mail. If not I will use such that, or
s.t., or things like that.
The expression {x ⎮ x is even} is literally read as: the set of
object x, (or number x if we are in a context where we talk about
number) such that x is even.
Exercise 1: Could you define in intension the following infinite set
C = {101, 103, 105, ...}
C = ?
Exercise 2: I will say that a natural number is a multiple of 4 if
it can be written as 4*y, for some y. For example 0 is a multiple of
4, (0 = 4*0), but also 28, 400, 404, ... Could you define in
extension the following set D = {x ⎮ x 10x is a multiple of
4}.
A last notational, but important symbol. Sets have elements. For
example the set A = {1, 2, 3} has three elements 1, 2 and 3. For
saying that 3 is an element of A in an a short way, we usually write
3 ∈ A. this is read as 3 belongs to A, or 3 is in A. Now 4
does not belong to A. To write this in a short way, we will write 4
∉ A, or we will write ¬ (4 ∈ A) or sometimes just NOT(4 ∈ A).
It is read: 4 does not belong to A, or: it is not the case that 4
belongs to A.
Having those notions and notations at our disposition we can speed
up on the notion of union and intersection.
The intersection of the sets A and B is the (new) set of those
elements which belongs to both A and B. Put in another way:
The intersection of the sets A with the set B is the set of those
elements which belongs to A and which belongs to B.
This new set, obtained from A and B is written A ∩ B, or A inter. B
(in case the special character doesn't go through).
With our notations we can write or define the intersection A ∩ B
directly
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B,
written A ∪ B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual
logical or. p or q is suppose to be true if p is true or q is true
(or both are true). It is not the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}.
Exercice 3.
Let N = {0, 1, 2, 3, ...}
Let A = {x ⎮ x 10}
Let B = {x ⎮ x
Re: The seven step-Mathematical preliminaries 2
On this date, you made the following correction: You cannot write D = 4*x
..., But you wrote D= 4*x in the exercise just above it. I don't get
the distinction between your use of the equation and mine.
- Original Message -
From: Bruno Marchal
Exercise 2: I will say that a natural number is a multiple of 4 if it can
be written as 4*y, for some y. For example 0 is a multiple of 4, (0 = 4*0), but
also 28, 400, 404, ... Could you define in extension the following set D = {x
⎮ x 10x is a multiple of 4}.D=4*x where x = 0 (but also 1,2,3...10)
You cannot write D = 4*x ..., given that D is a set, and 4*x is a (unknown)
number (a multiple of four when x is a natural number).
Read carefully the problem. I gave the set in intension, and the exercise
consisted in writing the set in extension. Let us translate in english the
definition of the set D = {x ⎮ x 10x is a multiple of 4}: it means that
D is the set of numbers, x, such that x is little than 10, and x is a multiple
of four. So D = {0, 4, 8}.
SEE BELOW
Example: the set of multiple of 4 is {0, 4, 8, 12, 16, 20, 24, 28, 32, 36,
...}, all have the shape 4*x, with x = to 0, 1, 2, 3, ...
The set of multiple of 5 is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55,
...}
Etc.
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B, written
A ∪ B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual logical or. p or
q is suppose to be true if p is true or q is true (or both are true). It is not
the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}. Question:
In the example above, 5,6 were the intersection because they were the (only)
two numbers BOTH groups had in common. But in this example, 7 is only in the
second group yet it is included in the answer. Please explain.
In the example above (that is {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}) we
were taking the INTERSECTION of the two sets.
But after that, may be too quickly (and I should have made a title perhaps) I
was introducing the UNION of the two sets.
If you read carefully the definition in intension, you should see that the
intersection of A and B is defined with an and. The definition of union is
defined with a or. Do you see that? It is just above in the quote.
I hope that your computer can distinguish A ∩ B (A intersection B) and A ∪ B
(A union B).
In the union of two sets, you put all the elements of the two sets together.
In the intersection of two sets, you take only those elements which belongs to
the two sets.
It seems you have not seen the difference between intersection and union.
This has indeed been the case. My usual math disabilities have been
exacerbated by the confusion of symbols due to E-mail limitations. The
profusion of little rectangles replacing the UNION symbol make the formulae
difficult to follow.
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Re: The seven step-Mathematical preliminaries 2
m.a. wrote:
*Bruno,*
* I've encountered some difficulty with the examples below.
You say that in extension describes exhaustion or
quasi-exhaustion. And you give the example: **B = {3, 6, 9, 12, ...
99}.*
* Then you define in intension with exactly the same type
of set: Example: Let A be the set {2, 4, 6, 8, 10, ... 100}.*
No, that's not the intensional definition. This We can easily define A
in intension: A = the set of numbers which are even and more little
than 100. is the intensional definition.
Brent
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Re: The seven step-Mathematical preliminaries 2
Bruno, When I tried to copy the symbols from the URL cited below, I found that my email server was not able to reproproduce the intersection or the union symbol. See below: From: Bruno Marchal To: everything-list@googlegroups.com ∅ ∪ A = I see two rectangles and A ∅ ∪ B = I see two rectangles and B A ∪ ∅ = I see A and two rectangles B ∪ ∅ = I see B and two rectangles N ∩ ∅ = I see N Inverted U and a rectangle B ∩ ∅ = I see B Inverted U and a rectangle ∅ ∩ B = I see a rectangle an inverted U and B ∅ ∩ ∅ = I see a rectangle an inverted U and a rectangle ∅ ∪ ∅ = I see three rectangles - Original Message - From: Bruno Marchal You could look on the archive, from here, http://www.mail-archive.com/everything-list@googlegroups.com/msg16531.html --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Quentin Anciaux wrote: 2009/6/6 Torgny Tholerus tor...@dsv.su.se: Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. What is the last number human can invent ? Your theory can't explain why addition works... If N is limited, then addition can and will (in human lifetime) create number which are still finite and not in N. It is very unlikely that anyone will get to the number 10^10^100 by addition. :-) Would agree that a any given time there is a largest number which has been conceived by a human being? N can be defined solelly as the successor function, you don't need anything else. You just have to assert that the function is true always. Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. Prove it please. That would seem to turn on the meaning of possible. Many (dare I say infinitely many) things are logically possible which are not nomologically possible (although the posters on this list seem to doubt that). Brent --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Okay, so is it true to say that things written in EXTENSION are never in
formula style but are translated into formulas when we put them into INTENSION
form? You can see that my difficulty with math arises from an inability to
master even the simplest definitions.marty a.
- Original Message -
From: Bruno Marchal
I've encountered some difficulty with the examples below. You
say that in extension describes exhaustion or quasi-exhaustion. And you
give the example: B = {3, 6, 9, 12, ... 99}.
Then you define in intension with exactly the same type of
set: Example: Let A be the set {2, 4, 6, 8, 10, ... 100}.
I give A in extension there, but just to define it in intension after. It is
always the same set there. But I show its definition in extension, to show the
definition in intension after. You have to read the to sentences.
Can you see the cause of my confusion?
It is always the same set. I give it in extension, and then in intension.
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
m.a. wrote: *Okay, so is it true to say that things written in EXTENSION are never in formula style but are translated into formulas when we put them into INTENSION form? You can see that my difficulty with math arises from an inability to master even the simplest definitions. marty a.* It's not that technical. I could define the set of books on my shelf by giving a list of titles: The Comprehensible Cosmos, Set Theory and It's Philosophy, Overshoot, Quintessence. That would be a definition by extension. Or I could point to them in succession and say, That and that and that and that. which would be a definition by ostension. Or I could just say, The books on my shelf. which is a definition by intension. An intensional definition is a descriptive phrase with an implicit variable, which in logic you might write as: The set of things x such that x is a book and x is on my shelf. Brent --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus tor...@dsv.su.se
mailto:tor...@dsv.su.se wrote:
Brian Tenneson skrev:
Torgny Tholerus wrote:
It is impossible to create a set where the successor of every
element is
inside the set, there must always be an element where the
successor of
that element is outside the set.
I disagree. Can you prove this?
Once again, I think the debate ultimately is about whether or not to
adopt the axiom of infinity.
I think everyone can agree without that axiom, you cannot build or
construct an infinite set.
There's nothing right or wrong with adopting any axioms. What
results
is either interesting or not, relevant or not.
How do you handle the Russell paradox with the set of all sets
that does
not contain itself? Does that set contain itself or not?
If we're talking about ZFC set theory, then the axiom of foundation
prohibits sets from being elements of themselves.
I think we agree that in ZFC, there is no set of all sets.
But there is a set of all sets. You can construct it by taking all
sets, and from them doing a new set, the set of all sets. But note,
this set will not contain itself, because that set did not exist before.
My answer is that that set does not contain itself, because no set can
contain itself. So the set of all sets that does not contain
itself, is
the same as the set of all sets. And that set does not contain
itself.
This set is a set, but it does not contain itself. It is exactly the
same with the natural numbers, *BIGGEST+1 is a natural number, but it
does not belong to the set of all natural numbers. *The set of
all sets
is a set, but it does not belong to the set of all sets.
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0, 1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers. BIGGEST+1 is bigger than
all natural numbers, because all natural numbers belongs to BIGGEST+1.
What the largest number is depends on how you define natural
number.
One possible definition is that N contains all explicit numbers
expressed by a human being, or will be expressed by a human
being in the
future. Amongst all those explicit numbers there will be one
that is
the largest. But this largest number is not an explicit number.
This raises a deeper question which is this: is mathematics
dependent
on humanity or is mathematics independent of humanity?
I wonder what would happen to that human being who finally expresses
the largest number in the future. What happens to him when he wakes
up the next day and considers adding one to yesterday's number?
This is no problem. If he adds one to the explicit number he
expressed
yesterday, then this new number is an explicit number, and the number
expressed yesterday was not the largest number. Both 17 and 17+1 are
explicit numbers.
This goes back to my earlier comment that it's hard for me to believe
that the following statement is false:
every natural number has a natural number successor
We -must- be talking about different things, then, when we use the
phrase natural number.
I can't say your definition of natural numbers is right and mine is
wrong, or vice versa. I do wonder what advantages there are to the
ultrafinitist approach compared to the math I'm familiar with.
The biggest advantage is that everything is finite, and you can then
really know that the mathematical theory you get is consistent, it does
not contain any contradictions.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
Kory Heath skrev: On Jun 4, 2009, at 8:27 AM, Torgny Tholerus wrote: How do you handle the Russell paradox with the set of all sets that does not contain itself? Does that set contain itself or not? My answer is that that set does not contain itself, because no set can contain itself. So the set of all sets that does not contain itself, is the same as the set of all sets. And that set does not contain itself. This set is a set, but it does not contain itself. It is exactly the same with the natural numbers, BIGGEST+1 is a natural number, but it does not belong to the set of all natural numbers. The set of all sets is a set, but it does not belong to the set of all sets. So you're saying that the set of all sets doesn't contain all sets. How is that any less paradoxical than the Russell paradox you're trying to avoid? The secret is the little word all. To be able to use that word, you have to define it. You can define it by saying: By 'all sets' I mean that set and that set and that set and When you have made that definition, you are then able to create a new set, the set of all sets. But you must be carefull with what you do with that set. That set does not contain itself, because it was not included in your definition of all sets. If you call the set of all sets for A, then you have: For all x such that x is a set, then x belongs to A. A is a set. But it is illegal to substitute A for x, so you can not deduce: A is a set, then A belongs to A. This deductuion is illegal, because A is not included in the definition of all x. -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/5 Torgny Tholerus tor...@dsv.su.se: Kory Heath skrev: On Jun 4, 2009, at 8:27 AM, Torgny Tholerus wrote: How do you handle the Russell paradox with the set of all sets that does not contain itself? Does that set contain itself or not? My answer is that that set does not contain itself, because no set can contain itself. So the set of all sets that does not contain itself, is the same as the set of all sets. And that set does not contain itself. This set is a set, but it does not contain itself. It is exactly the same with the natural numbers, BIGGEST+1 is a natural number, but it does not belong to the set of all natural numbers. The set of all sets is a set, but it does not belong to the set of all sets. So you're saying that the set of all sets doesn't contain all sets. How is that any less paradoxical than the Russell paradox you're trying to avoid? The secret is the little word all. To be able to use that word, you have to define it. I call that secret bullshit, and to understand that word (bullshit), you have to define it. Sorry but I think we're talking in english here, all means all not what you decide it means. Quentin. You can define it by saying: By 'all sets' I mean that set and that set and that set and When you have made that definition, you are then able to create a new set, the set of all sets. But you must be carefull with what you do with that set. That set does not contain itself, because it was not included in your definition of all sets. If you call the set of all sets for A, then you have: For all x such that x is a set, then x belongs to A. A is a set. But it is illegal to substitute A for x, so you can not deduce: A is a set, then A belongs to A. This deductuion is illegal, because A is not included in the definition of all x. -- Torgny Tholerus -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Fri, 5 Jun 2009 08:33:47 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus tor...@dsv.su.se
mailto:tor...@dsv.su.se wrote:
Brian Tenneson skrev:
Torgny Tholerus wrote:
It is impossible to create a set where the successor of every
element is
inside the set, there must always be an element where the
successor of
that element is outside the set.
I disagree. Can you prove this?
Once again, I think the debate ultimately is about whether or not to
adopt the axiom of infinity.
I think everyone can agree without that axiom, you cannot build or
construct an infinite set.
There's nothing right or wrong with adopting any axioms. What
results
is either interesting or not, relevant or not.
How do you handle the Russell paradox with the set of all sets
that does
not contain itself? Does that set contain itself or not?
If we're talking about ZFC set theory, then the axiom of foundation
prohibits sets from being elements of themselves.
I think we agree that in ZFC, there is no set of all sets.
But there is a set of all sets. You can construct it by taking all
sets, and from them doing a new set, the set of all sets. But note,
this set will not contain itself, because that set did not exist before.
My answer is that that set does not contain itself, because no set can
contain itself. So the set of all sets that does not contain
itself, is
the same as the set of all sets. And that set does not contain
itself.
This set is a set, but it does not contain itself. It is exactly the
same with the natural numbers, *BIGGEST+1 is a natural number, but it
does not belong to the set of all natural numbers. *The set of
all sets
is a set, but it does not belong to the set of all sets.
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0, 1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers.
Here you're just contradicting yourself. If you say BIGGEST+1 is then a
natural number, that just proves that the set N was not in fact the set of
all natural numbers. The alternative would be to say BIGGEST+1 is *not* a
natural number, but then you need to provide a definition of natural number
that would explain why this is the case.
The biggest advantage is that everything is finite, and you can then
really know that the mathematical theory you get is consistent, it does
not contain any contradictions.
Even if you define natural number in such a way that there are only a finite
number of them (which you haven't actually done, you've just asserted it
without providing any specific definition), you still could have an infinite
number of *propositions* about them if you allow each proposition to contain an
unlimited number of AND and OR operators. For example, even if I say that the
only natural numbers are 1,2,3, I can still make arbitrarily long propositions
like ((31) AND (21)) OR (31)) AND ((23) OR (31)) AND ((23) OR ((13) OR
((21) OR ((13) OR (31). Of course a non-finitist would be able to prove
that these infinite number of propositions are consistent, but I don't know if
an ultrafinitist would (likewise a non-finitist can accept a proof that
something like the Peano axioms are consistent based on an understanding of
their application to a model dealing with rows of dots, even if the Peano
axioms cannot be used to formally prove their own consistency).
Jesse
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Re: The seven step-Mathematical preliminaries
Torgny Tholerus wrote:
Brian Tenneson skrev:
On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus tor...@dsv.su.se
mailto:tor...@dsv.su.se wrote:
Brian Tenneson skrev:
Torgny Tholerus wrote:
It is impossible to create a set where the successor of every
element is
inside the set, there must always be an element where the
successor of
that element is outside the set.
I disagree. Can you prove this?
Once again, I think the debate ultimately is about whether or not to
adopt the axiom of infinity.
I think everyone can agree without that axiom, you cannot build or
construct an infinite set.
There's nothing right or wrong with adopting any axioms. What
results
is either interesting or not, relevant or not.
How do you handle the Russell paradox with the set of all sets
that does
not contain itself? Does that set contain itself or not?
If we're talking about ZFC set theory, then the axiom of foundation
prohibits sets from being elements of themselves.
I think we agree that in ZFC, there is no set of all sets.
But there is a set of all sets. You can construct it by taking all
sets, and from them doing a new set, the set of all sets. But note,
this set will not contain itself, because that set did not exist before.
If that set does not contain itself then it is not a set of all sets.
My answer is that that set does not contain itself, because no set can
contain itself. So the set of all sets that does not contain
itself, is
the same as the set of all sets. And that set does not contain
itself.
This set is a set, but it does not contain itself. It is exactly the
same with the natural numbers, *BIGGEST+1 is a natural number, but it
does not belong to the set of all natural numbers. *The set of
all sets
is a set, but it does not belong to the set of all sets.
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0, 1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers. BIGGEST+1 is bigger than
all natural numbers, because all natural numbers belongs to BIGGEST+1.
Right, so n+1 is a natural number whenever n is.
What the largest number is depends on how you define natural
number.
One possible definition is that N contains all explicit numbers
expressed by a human being, or will be expressed by a human
being in the
future. Amongst all those explicit numbers there will be one
that is
the largest. But this largest number is not an explicit number.
This raises a deeper question which is this: is mathematics
dependent
on humanity or is mathematics independent of humanity?
I wonder what would happen to that human being who finally expresses
the largest number in the future. What happens to him when he wakes
up the next day and considers adding one to yesterday's number?
This is no problem. If he adds one to the explicit number he
expressed
yesterday, then this new number is an explicit number, and the number
expressed yesterday was not the largest number. Both 17 and 17+1 are
explicit numbers.
This goes back to my earlier comment that it's hard for me to believe
that the following statement is false:
every natural number has a natural number successor
We -must- be talking about different things, then, when we use the
phrase natural number.
I can't say your definition of natural numbers is right and mine is
wrong, or vice versa. I do wonder what advantages there are to the
ultrafinitist approach compared to the math I'm familiar with.
The biggest advantage is that everything is finite, and you can then
really know that the mathematical theory you get is consistent, it does
not contain any contradictions.
From what you said earlier, BIGGEST={0,1,...,BIGGEST-1}. Then
BIGGEST+1={0,1,...,BIGGEST-1} union {BIGGEST} = {0,1,...,BIGGEST}.
Why would {0,1,...BIGGEST} not be a natural number while
{0,1,...,BIGGEST-1} is?
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Re: The seven step-Mathematical preliminaries
From what you said earlier, BIGGEST={0,1,...,BIGGEST-1}. Then
BIGGEST+1={0,1,...,BIGGEST-1} union {BIGGEST} = {0,1,...,BIGGEST}.
Why would {0,1,...BIGGEST} not be a natural number while
{0,1,...,BIGGEST-1} is?
If {0, 1, ... , BIGGEST-1} is a natural number, then {0,1,...,BIGGEST} is
too, and then so is {0, 1, ... , BIGGEST+1}, etc. There's no such thing as
a largest natural number: that's the whole point of the construction. The
set of all natural numbers is an infinite set, unbounded above. The set N
has no largest element within it: it is the set of all finite ordinals. N
(usually called omega when treated as an ordinal) has no predecessor,
because it is formed by taking the limit of all the ordinals below it, *not*
by applying the successor function x+ = x U {x}. This is the way
well-ordering works...it's not symmetric. So any set described {a, b, ... ,
z} in the standard way is not N.
N is not the successor of any natural number; rather, it contains them all.
This allows us to talk about (and prove things about) all natural numbers.
This isn't an arbitrary mathematical choice. Without infinite sets, we
would be unable to rigorously prove things by induction, which is necessary
for a wide array of basic arithmetical proofs. This is because a finite set
of natural numbers cannot be closed under successor (or addition or
multiplication, for that matter). If you relied on only finitely many
numbers, your functions could take natural numbers and hand you back
something that isn't a number at all. This makes even basic math untenable.
Taking the closure of {} under successor is the solution.
(There are non-standard models of the natural numbers that contain numbers
other than the elements of N, but these are not well-ordered.)
Anna
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Re: The seven step-Mathematical preliminaries 2
Bruno, Thanks for the encouragement. I intend to follow your instructions and it's a relief to know that some of my answers were correct. However, I will be away for two weeks and unable to work on the lessons. I'll try to make up for it when I return. Best, marty a. - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Friday, June 05, 2009 10:03 AM Subject: Re: The seven step-Mathematical preliminaries 2 Hi Marty, On 05 Jun 2009, at 00:30, m.a. wrote: Bruno, I don't have dyslexia Good news. but my keyboard doesn't contain either the UNION symbol or the INTERSECTION symbol Nor do mine! (unless I want to go into an INSERT pull down menu every time I use those symbols). Like I have to do too. I don't need you to switch to English symbols, but I would like to see the English equivalents of the symbols you use (so that I can use them). I gave them. I would also like a reference table defining each term in both your symbols and their English equivalents which I could look back to when I get confused. I suggest you do this by yourself. It is a good exercise and it will help you not only in the understanding, but in the memorizing. Then you submit it to the list, and I can verify the understanding. Please include examples. Up to now, I did it for any notions introduced. Just ask me one or two or (name your number) examples more in case you have a doubt. If I send too much posts, and if there are too long, people will dismiss them. try to ask explicit question, like you did, actually. I tend to be somewhat careless when dealing with very fine distinctions This means that a lot of work is awaiting for you. It is normal. Everyone can understand what I explain, but some have more work to do. and may type the wrong symbol while intending to type the correct one. That is unimportant. I am used to do typo errors too. One of my favorite book on self-reference (the one by Smorynski) contains an average of two or three typo error per page. Of course, once a typo error is found, it is better to correct it. Also, I must admit that the lessons are going too fast for me and are moving ahead before I've mastered the previous material. We have all the time, and up to now I did not proceed without having the answer of all exercises. You make no faults in the first set of seven exercise, and that is why I have quickly proceed to the second round. For that one, you make just one error, + the dismiss of a paragraph on UNION. To slow me down it is enough to tell me things like I don't understand what you mean by this or that and you quote the unclear passage. If you can't do an exercise, just wait for some other (Kim?) to propose a solution. Or try to guess one and submit, or just ask. I will not proceed to new matters before I am sure you grasp all what has been already presented. What is possible is that you understand, but fail ti memorize. This will lead to problems later. So you have to make your own summary and be sure you can easily revise the definition. If I'm requesting too much simplification, please let me know because I'm quite well adjusted to my math disabilities and won't take offence at all. Thanks, marty a. I think that there is no problem at all. I am just waiting for explicit question from the second round. You can ask any question, and slow me down as much as you want so that we proceed at your own rhythm. Don't ask me to slow down in any abstract way. You are the one who have to slow me down by pointing on what you don't understand in a post. take it easy, and take all your time. Don't try to understand the more advanced replies I give to people who have a bigger baggage. You did show me that you have understood the notion of set, and the notion of intersection of sets. Have you a problem with the notion of union of sets? If that is the case, just quote the passage of my post that you don't understand, or the example that I gave, and I will explain. Try to keep those post in some well ranged place so as to re-access them easily. I ask this to Kim too, and any one interested: just let me know what you don't understand, so that I can explain, give other examples, etc. Take it easy, you seem quite good, you suffer just of a problem of familiarity with notations. You read the post too quickly, I suspect also. Are you OK? I can understand you could be afraid of the amount of work, but given that we have all the time, there is no exams, nor deadline, I am not sure there is any problem
Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
How do you know that there is no biggest number? Have you examined all
the natural numbers? How do you prove that there is no biggest number?
In my opinion those are excellent questions. I will attempt to answer
them. The intended audience of my answer is everyone, so please forgive
me if I say something you already know.
Firstly, no one has or can examine all the natural numbers. By that I
mean no human. Maybe there is an omniscient machine (or a maximally
knowledgeable in some paraconsistent way) who can examine all numbers
but that is definitely putting the cart before the horse.
Secondly, the question boils down to a difference in philosophy:
mathematicians would say that it is not necessary to examine all natural
numbers. The mathematician would argue that it suffices to examine all
essential properties of natural numbers, rather than all natural numbers.
There are a variety of equivalent ways to define a natural number but
the essential features of natural numbers are that
(a) there is an ordering on the set of natural numbers, a well
ordering. To say a set is well ordered entails that every =nonempty=
subset of it has a least element.
(b) the set of natural numbers has a least element (note that it is
customary to either say 0 is this least element or say 1 is this least
element--in some sense it does not matter what the starting point is)
(c) every natural number has a natural number successor. By successor
of a natural number, I mean anything for which the well ordering always
places the successor as larger than the predecessor.
Then the set of natural numbers, N, is the set containing the least
element (0 or 1) and every successor of the least element, and only
successors of the least element.
There is nothing wrong with a proof by contradiction but I think a
forward proof might just be more convincing.
Consider the following statement:
Whenever S is a subset of N, S has a largest element if, and only if,
the complement of S has a least element.
By complement of S, I mean the set of all elements of N that are not
elements of S.
Before I give a longer argument, would you agree that statement is
true? One can actually be arbitrarily explicit: M is the largest
element of S if, and only if, the successor of M is the least element of
the compliment of S.
I do not agree that statement is true. Because if you call the Biggest
natural number B, then you can describe N as = {1, 2, 3, ..., B}. If
you take the complement of N you will get the empty set. This set have
no least element, but still N has a biggest element.
In your statement you are presupposing that N has no biggest element,
and from that axiom you can trivially deduce that there is no biggest
element.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
If you are ultrafinitist then by definition the set N does not
exist... (nor any infinite set countably or not).
If you pose the assumption of a biggest number for N, you come to a
contradiction because you use the successor operation which cannot
admit a biggest number.(because N is well ordered any successor is
strictly bigger and the successor operation is always valid *by
definition of the operation*)
So either the set N does not exists in which case it makes no sense to
talk about the biggest number in N, or the set N does indeed exists
and it makes no sense to talk about the biggest number in N (while it
makes sense to talk about a number which is strictly bigger than any
natural number).
To come back to the proof by contradiction you gave, the assumption
(2) which is that BIGGEST+1 is in N, is completely defined by the mere
existence of BIGGEST. If BIGGEST exists and well defined it entails
that BIGGEST+1 is not in N (but this invalidate the successor
operation and hence the mere existence of N). If BIGGEST in contrary
does not exist (as such, means it is not the biggest) then BIGGEST+1
is in N by definition of N.
Regards,
Quentin
2009/6/4 Torgny Tholerus tor...@dsv.su.se:
Brian Tenneson skrev:
How do you know that there is no biggest number? Have you examined all
the natural numbers? How do you prove that there is no biggest number?
In my opinion those are excellent questions. I will attempt to answer
them. The intended audience of my answer is everyone, so please forgive
me if I say something you already know.
Firstly, no one has or can examine all the natural numbers. By that I
mean no human. Maybe there is an omniscient machine (or a maximally
knowledgeable in some paraconsistent way) who can examine all numbers
but that is definitely putting the cart before the horse.
Secondly, the question boils down to a difference in philosophy:
mathematicians would say that it is not necessary to examine all natural
numbers. The mathematician would argue that it suffices to examine all
essential properties of natural numbers, rather than all natural numbers.
There are a variety of equivalent ways to define a natural number but
the essential features of natural numbers are that
(a) there is an ordering on the set of natural numbers, a well
ordering. To say a set is well ordered entails that every =nonempty=
subset of it has a least element.
(b) the set of natural numbers has a least element (note that it is
customary to either say 0 is this least element or say 1 is this least
element--in some sense it does not matter what the starting point is)
(c) every natural number has a natural number successor. By successor
of a natural number, I mean anything for which the well ordering always
places the successor as larger than the predecessor.
Then the set of natural numbers, N, is the set containing the least
element (0 or 1) and every successor of the least element, and only
successors of the least element.
There is nothing wrong with a proof by contradiction but I think a
forward proof might just be more convincing.
Consider the following statement:
Whenever S is a subset of N, S has a largest element if, and only if,
the complement of S has a least element.
By complement of S, I mean the set of all elements of N that are not
elements of S.
Before I give a longer argument, would you agree that statement is
true? One can actually be arbitrarily explicit: M is the largest
element of S if, and only if, the successor of M is the least element of
the compliment of S.
I do not agree that statement is true. Because if you call the Biggest
natural number B, then you can describe N as = {1, 2, 3, ..., B}. If
you take the complement of N you will get the empty set. This set have
no least element, but still N has a biggest element.
In your statement you are presupposing that N has no biggest element,
and from that axiom you can trivially deduce that there is no biggest
element.
--
Torgny Tholerus
--
All those moments will be lost in time, like tears in rain.
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Re: The seven step-Mathematical preliminaries 2
On Thu Jun 4 1:15 , Bruno Marchal sent:
Very good answer, Kim,
Just a few comments. and then the sequel.
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
No idea what square-root(2) means. When I said I was innumerate I wasn't
kidding! I
could of course look
it up or ask my mathematics teacher friends but I just know your explanation
will make
theirs seem trite.
Well thanks. The square root of 2 is a number x, such that x*x (x times x, x
multiplied by
itself) gives 2.For example, the square root of 4 is 2, because 2*2 is 4. The
square root of
9 is 3, because 3*3 is 9. Her by square root I mean the positive square root,
because we
will see (more later that soon) that numbers can have negative square root, but
please
forget this. At this stage, with this definition, you can guess that the square
root of 2
cannot be a natural number. 1*1 = 1, and 2*2 = 4, and it would be astonishing
that x
could be bigger than 2. So if there is number x such that x*x is 2, we can
guess that such
a x cannot be a natural number, that is an element of {0, 1, 2, 3 ...}, and the
answer of
exercise 4 is no. The square root of two will reappear recurrently, but more
in examples,
than in the sequence of notions which are strictly needed for UDA-7.
OK - I find this quite mind-blowing; probably because I now understand it for
the first
time in my life. So how did it get this rather ridiculous name of square
root? What's it
called in French?
(snip)
=== Intension and extension
Before defining intersection, union and the notion of subset, I would like to
come back
on the ways we can define some specific sets.
In the case of finite and little set we have seen that we can define them by
exhaustion.
This means we can give an explicit complete description of all element of the
set. Example. A = {0, 1, 2, 77, 98, 5}
When the set is still finite and too big, or if we are lazy, we can sometimes
define the set
by quasi exhaustion. This means we describe enough elements of the set in a
manner
which, by requiring some good will and some imagination, we can estimate having
define
the set.
Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case that we
meant the set of
multiple of the number three, below 100.
A fortiori, when a set in not finite, that is, when the set is infinite, we
have to use either
quasi-exhaustion, or we have to use some sentence or phrase or proposition
describing
the elements of the set.
Definition. I will say that a set is defined IN EXTENSIO, or simply, in
extension, when it is
defined in exhaustion or quasi-exhaustion. I will say that a set is defined IN
INTENSIO, or
simply in intension, with an s, when it is defined by a sentence explaining
the typical
attribute of the elements.
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily define A in
intension: A
= the set of numbers which are even and smaller than 100. Mathematicians will
condense
this by the following:
A = {x such that x is even and smaller than 100} = {x ⎮ x is even x
special character, abbreviating such that, and I hope it goes through the
mail.
Just an upright line? It comes through as that. I can't seem to get this symbol
happening so I will
use such that
If not I will use such that, or s.t., or things like that.The expression
{x ⎮ x is even} is
literally read as: the set of objects x, (or number x if we are in a context
where we talk
about numbers) such that x is even.
Exercise 1: Could you define in intension the following infinite set C = {101,
103, 105,
...}C = ?
C = {x such that x is odd and x 101}
Exercise 2: I will say that a natural number is a multiple of 4 if it can be
written as 4*y,
for some y. For example 0 is a multiple of 4, (0 = 4*0), but also 28, 400, 404,
... Could
you define in extension the following set D = {x ⎮ x 10 and x is a multiple
of 4}?
D = 4*x where x = 0 but also { 1, 2, 3, 4, 8 }
I now realise I am doomed for the next set of exercises because I cannot get to
the special
symbols required (yet). As I am adding Internet Phone to my system, I am
currently using an
ancient Mac without the correct symbol pallette while somebody spends a few
days to flip a single
switch...as soon as I can get back to my regular machine I will complete the
rest.
In the meantime I am enjoying the N+1 disagreement - how refreshing it is to
see that classical
mathematics remains somewhat controversial!
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Re: The seven step-Mathematical preliminaries
Quentin Anciaux skrev: If you are ultrafinitist then by definition the set N does not exist... (nor any infinite set countably or not). All sets are finite. It it (logically) impossible to construct an infinite set. You can construct the set N of all natural numbers. But that set must be finite. What the set N contains depends on how you have defined natural number. If you pose the assumption of a biggest number for N, you come to a contradiction because you use the successor operation which cannot admit a biggest number.(because N is well ordered any successor is strictly bigger and the successor operation is always valid *by definition of the operation*) You have to define the successor operation. And to do that you have to define the definition set for that operation. So first you have to define the set N of natural numbers. And from that you can define the successor operator. The value set of the successor operator will be a new set, that contains one more element than the set N of natural numbers. This new element is BIGGEST+1, that is strictly bigger than all natural numbers. -- Torgny Tholerus So either the set N does not exists in which case it makes no sense to talk about the biggest number in N, or the set N does indeed exists and it makes no sense to talk about the biggest number in N (while it makes sense to talk about a number which is strictly bigger than any natural number). To come back to the proof by contradiction you gave, the assumption (2) which is that BIGGEST+1 is in N, is completely defined by the mere existence of BIGGEST. If BIGGEST exists and well defined it entails that BIGGEST+1 is not in N (but this invalidate the successor operation and hence the mere existence of N). If BIGGEST in contrary does not exist (as such, means it is not the biggest) then BIGGEST+1 is in N by definition of N. Regards, Quentin --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. For me it is strange that the following statement is false: every natural number has a natural number successor. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. Torgny Tholerus wrote: Quentin Anciaux skrev: If you are ultrafinitist then by definition the set N does not exist... (nor any infinite set countably or not). All sets are finite. It it (logically) impossible to construct an infinite set. You can construct the set N of all natural numbers. But that set must be finite. What the set N contains depends on how you have defined natural number. If you pose the assumption of a biggest number for N, you come to a contradiction because you use the successor operation which cannot admit a biggest number.(because N is well ordered any successor is strictly bigger and the successor operation is always valid *by definition of the operation*) You have to define the successor operation. And to do that you have to define the definition set for that operation. So first you have to define the set N of natural numbers. And from that you can define the successor operator. The value set of the successor operator will be a new set, that contains one more element than the set N of natural numbers. This new element is BIGGEST+1, that is strictly bigger than all natural numbers. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev: This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. For me it is strange that the following statement is false: every natural number has a natural number successor. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. It is impossible to lock a box, and quickly throw the key inside the box before you lock it. It is impossible to create a set and put the set itself inside the set, i.e. no set can contain itself. It is impossible to create a set where the successor of every element is inside the set, there must always be an element where the successor of that element is outside the set. What the largest number is depends on how you define natural number. One possible definition is that N contains all explicit numbers expressed by a human being, or will be expressed by a human being in the future. Amongst all those explicit numbers there will be one that is the largest. But this largest number is not an explicit number. -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Torgny Tholerus wrote: Brian Tenneson skrev: This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. For me it is strange that the following statement is false: every natural number has a natural number successor. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. It is impossible to lock a box, and quickly throw the key inside the box before you lock it. I disagree. It is impossible to create a set and put the set itself inside the set, i.e. no set can contain itself. No one here is suggesting that you can with regards to natural numbers. It is impossible to create a set where the successor of every element is inside the set, there must always be an element where the successor of that element is outside the set. I disagree. Can you prove this? Once again, I think the debate ultimately is about whether or not to adopt the axiom of infinity. I think everyone can agree without that axiom, you cannot build or construct an infinite set. There's nothing right or wrong with adopting any axioms. What results is either interesting or not, relevant or not. What the largest number is depends on how you define natural number. One possible definition is that N contains all explicit numbers expressed by a human being, or will be expressed by a human being in the future. Amongst all those explicit numbers there will be one that is the largest. But this largest number is not an explicit number. This raises a deeper question which is this: is mathematics dependent on humanity or is mathematics independent of humanity? I wonder what would happen to that human being who finally expresses the largest number in the future. What happens to him when he wakes up the next day and considers adding one to yesterday's number? --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Hi Marty,
On 04 Jun 2009, at 01:11, m.a. wrote:
Bruno,
I stopped half-way through because I'm not at all sure of
my answers and would like to have them confirmed or corrected, if
necessary, rather than go on giving wrong answers. marty a.
No problem.
Exercise 1: Could you define in intension the following infinite set
C = {101, 103, 105, ...}
C = ? C={x such that x is odd x 101}
I guess you meant C = {x such that x is odd and x 101}. means
bigger than, and means little than. OK.
Exercise 2: I will say that a natural number is a multiple of 4 if
it can be written as 4*y, for some y. For example 0 is a multiple of
4, (0 = 4*0), but also 28, 400, 404, ... Could you define in
extension the following set D = {x ⎮ x 10x is a multiple of
4}.D=4*x where x = 0 (but also 1,2,3...10)
You cannot write D = 4*x ..., given that D is a set, and 4*x is a
(unknown) number (a multiple of four when x is a natural number).
Read carefully the problem. I gave the set in intension, and the
exercise consisted in writing the set in extension. Let us translate
in english the definition of the set D = {x ⎮ x 10x is a
multiple of 4}: it means that D is the set of numbers, x, such that x
is little than 10, and x is a multiple of four. So D = {0, 4, 8}.
Indeed 0 is little than 10, and 0 is a multiple of four (because 0 =
4*0), and
4 is little than 10, and 4 is a multiple of four (because 4 = 4*1)
8 is little than 10, and 8 is a multiple of 4 (because 8 = 4*2)
The next mutiple of 4 is 12. It cannot be in the set because 12 is
bigger than 10.
The numbers 1, 2, 3, 5, 7, 9 cannot be in D, because they are not
multiple of 4. You cannot write 1 = 4 * (some natural numbers), nor
can you write 3 or 5, or 7 or 9 = 4 * x with x a natural number.
Example: the set of multiple of 4 is {0, 4, 8, 12, 16, 20, 24, 28, 32,
36, ...}, all have the shape 4*x, with x = to 0, 1, 2, 3, ...
The set of multiple of 5 is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50,
55, ...}
Etc.
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B,
written A ∪ B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual
logical or. p or q is suppose to be true if p is true or q is true
(or both are true). It is not the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}.
Question: In the example above, 5,6 were the intersection because
they were the (only) two numbers BOTH groups had in common. But in
this example, 7 is only in the second group yet it is included in
the answer. Please explain.
In the example above (that is {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5,
6}) we were taking the INTERSECTION of the two sets.
But after that, may be too quickly (and I should have made a title
perhaps) I was introducing the UNION of the two sets.
If you read carefully the definition in intension, you should see that
the intersection of A and B is defined with an and. The definition
of union is defined with a or. Do you see that? It is just above in
the quote.
I hope that your computer can distinguish A ∩ B (A intersection B)
and A ∪ B (A union B).
In the union of two sets, you put all the elements of the two sets
together. In the intersection of two sets, you take only those
elements which belongs to the two sets.
It seems you have not seen the difference between intersection and
union. I guess you try to go to much quickly, or that the font of
your computer are too little, or that you have eyesight problems, or
that you have some dyslexia.
Exercice 3.
Let N = {0, 1, 2, 3, ...}
Let A = {x ⎮ x 10}
Let B = {x ⎮ x is even}
Describe in extension (that is: exhaustion or quasi-exhaustion) the
following sets:
N ∪ A = {0,1,2,3...} inter {x inter x10}= {0,1,2,3...9}
N ∪ B = {0,1,2,3} inter {x inter x is even}= {0,2,4,6...}
A ∪ B = {x inter x 10} inter {x inter x is even}= {0,2,4,6,8}
B ∪ A = {x inter x is even} inter {x inter x 10}= {0,2,4,6,8}
All that would be correct if you were searching the intersection, but
∪ is the UNION symbol. (and ∩ is the INTERSECTION symbol).
also you wrote the ⎮ as inter, instead of such that.
N ∩ A = {0,1,2,3...} inter {x inter x10}= {0,1,2,3...9}
B ∩ A = {x inter x is even} inter {x inter x 10}= {0,2,4,6,8}
N ∩ B = {0,1,2,3} inter {x inter x is even}= {0,2,4,6...}
A ∩ B = {x inter x 10} inter {x inter x is even}= {0,2,4,6,8}
All that is correct. Careful you were still using inter in place of
such that. Your last line should be
A ∩ B = {x such that x 10} inter {x such that x is even}=
{0,2,4,6,8}
Exercice 4
Is it true that A ∩ B = B ∩ A, whatever A and B are? yes
Is it true that A ∪ B = B ∪ A, whatever A and B are? yes
Both are correct.
Not bad Marty! Just read carefully. I
Re: The seven step-Mathematical preliminaries 2
On Thu, Jun 4, 2009 at 7:28 AM, kimjo...@ozemail.com.au
kimjo...@ozemail.com.au wrote:
On Thu Jun 4 1:15 , Bruno Marchal sent:
Very good answer, Kim,
Just a few comments. and then the sequel.
Exercice 4: does the real number square-root(2) belongs to {0, 1, 2,
3, ...}?
No idea what square-root(2) means. When I said I was innumerate I wasn't
kidding! I
could of course look
it up or ask my mathematics teacher friends but I just know your explanation
will make
theirs seem trite.
Well thanks. The square root of 2 is a number x, such that x*x (x times x, x
multiplied by
itself) gives 2.For example, the square root of 4 is 2, because 2*2 is 4. The
square root of
9 is 3, because 3*3 is 9. Her by square root I mean the positive square
root, because we
will see (more later that soon) that numbers can have negative square root,
but please
forget this. At this stage, with this definition, you can guess that the
square root of 2
cannot be a natural number. 1*1 = 1, and 2*2 = 4, and it would be astonishing
that x
could be bigger than 2. So if there is number x such that x*x is 2, we can
guess that such
a x cannot be a natural number, that is an element of {0, 1, 2, 3 ...}, and
the answer of
exercise 4 is no. The square root of two will reappear recurrently, but
more in examples,
than in the sequence of notions which are strictly needed for UDA-7.
OK - I find this quite mind-blowing; probably because I now understand it for
the first
time in my life. So how did it get this rather ridiculous name of square
root? What's it
called in French?
I don't know what it is called in French, but I can answer the first
part. I remember the day I first figured out where the term came
from.
When you have a number multiplied by itself, the result is called a
square. 3*3 = 9, so 9 is a square. Imagine arranging a set of peas,
if you can arrange them in a square (the four cornered kind) with the
same number of rows as columns, then that number is a square. Some
examples of squares are: 4, 9, 16, 25, 36, 49, 64, 81, see the
pattern? And the roots of those squares are 2, 3, 4, 5, 6, 7, 8,
and 9. The square root is equal to the number of items in a row, or
column when you arrange them in a square.
This is a completely extraneous fact, but one I consider to be very
interesting: Multiply any 4 consecutive positive whole numbers and the
result will always be 1 less than a square number. For example,
5*6*7*8 = 1680, which is 1 less than 1681, which is 41*41. Isn't that
neat?
Jason
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Re: The seven step-Mathematical preliminaries
I've never seen an ultrafinitist definition of the natural numbers.
The usual definition via Peano's axioms obviously rules out there being
a largest number. I would suppose that an ultrafinitist definition of
the natural numbers would be something like seen in a computer (which is
necessarily finite). The successor operation would be defined such that
Successor (Biggest) = 0 or -Biggest.
Brent
Quentin Anciaux wrote:
If you are ultrafinitist then by definition the set N does not
exist... (nor any infinite set countably or not).
If you pose the assumption of a biggest number for N, you come to a
contradiction because you use the successor operation which cannot
admit a biggest number.(because N is well ordered any successor is
strictly bigger and the successor operation is always valid *by
definition of the operation*)
So either the set N does not exists in which case it makes no sense to
talk about the biggest number in N, or the set N does indeed exists
and it makes no sense to talk about the biggest number in N (while it
makes sense to talk about a number which is strictly bigger than any
natural number).
To come back to the proof by contradiction you gave, the assumption
(2) which is that BIGGEST+1 is in N, is completely defined by the mere
existence of BIGGEST. If BIGGEST exists and well defined it entails
that BIGGEST+1 is not in N (but this invalidate the successor
operation and hence the mere existence of N). If BIGGEST in contrary
does not exist (as such, means it is not the biggest) then BIGGEST+1
is in N by definition of N.
Regards,
Quentin
2009/6/4 Torgny Tholerus tor...@dsv.su.se:
Brian Tenneson skrev:
How do you know that there is no biggest number? Have you examined all
the natural numbers? How do you prove that there is no biggest number?
In my opinion those are excellent questions. I will attempt to answer
them. The intended audience of my answer is everyone, so please forgive
me if I say something you already know.
Firstly, no one has or can examine all the natural numbers. By that I
mean no human. Maybe there is an omniscient machine (or a maximally
knowledgeable in some paraconsistent way) who can examine all numbers
but that is definitely putting the cart before the horse.
Secondly, the question boils down to a difference in philosophy:
mathematicians would say that it is not necessary to examine all natural
numbers. The mathematician would argue that it suffices to examine all
essential properties of natural numbers, rather than all natural numbers.
There are a variety of equivalent ways to define a natural number but
the essential features of natural numbers are that
(a) there is an ordering on the set of natural numbers, a well
ordering. To say a set is well ordered entails that every =nonempty=
subset of it has a least element.
(b) the set of natural numbers has a least element (note that it is
customary to either say 0 is this least element or say 1 is this least
element--in some sense it does not matter what the starting point is)
(c) every natural number has a natural number successor. By successor
of a natural number, I mean anything for which the well ordering always
places the successor as larger than the predecessor.
Then the set of natural numbers, N, is the set containing the least
element (0 or 1) and every successor of the least element, and only
successors of the least element.
There is nothing wrong with a proof by contradiction but I think a
forward proof might just be more convincing.
Consider the following statement:
Whenever S is a subset of N, S has a largest element if, and only if,
the complement of S has a least element.
By complement of S, I mean the set of all elements of N that are not
elements of S.
Before I give a longer argument, would you agree that statement is
true? One can actually be arbitrarily explicit: M is the largest
element of S if, and only if, the successor of M is the least element of
the compliment of S.
I do not agree that statement is true. Because if you call the Biggest
natural number B, then you can describe N as = {1, 2, 3, ..., B}. If
you take the complement of N you will get the empty set. This set have
no least element, but still N has a biggest element.
In your statement you are presupposing that N has no biggest element,
and from that axiom you can trivially deduce that there is no biggest
element.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
Torgny Tholerus wrote: Brian Tenneson skrev: This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. For me it is strange that the following statement is false: every natural number has a natural number successor. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. It is impossible to lock a box, and quickly throw the key inside the box before you lock it. It is impossible to create a set and put the set itself inside the set, i.e. no set can contain itself. It is impossible to create a set where the successor of every element is inside the set, there must always be an element where the successor of that element is outside the set. Depends on how you define successor. Brent What the largest number is depends on how you define natural number. One possible definition is that N contains all explicit numbers expressed by a human being, or will be expressed by a human being in the future. Amongst all those explicit numbers there will be one that is the largest. But this largest number is not an explicit number. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
On 04 Jun 2009, at 15:40, Brian Tenneson wrote: This is a denial of the axiom of infinity. I think a foundational set theorist might agree that it is impossible to -construct- an infinite set from scratch which is why they use the axiom of infinity. People are free to deny axioms, of course, though the result will not be like ZFC set theory. The denial of axiom of foundation is one I've come across; I've never met anyone who denies the axiom of infinity. Among mathematicians nobody denies the axiom of infinity, but many philosopher of mathematics are attracted by finitism. But Torgny is ultrafinitist. That is much rare. he denies the existence of natural numbers above some rather putative biggest natural number. For me it is strange that the following statement is false: every natural number has a natural number successor. I thought he would have said this, and accepted that the successor of its N is equal to N+1. Nut in a reply he says that N+1 exists but is not a natural number, which I think should not be consistent. To me it seems quite arbitrary for the ultrafinitist's statement: every natural number has a natural number successor UNTIL we reach some natural number which does not have a natural number successor. I'm left wondering what the largest ultrafinist's number is. It cannot be a constructive object. It is a number which is so big that if you add 1 to it, the everything explodes! I dunno. I still suspect that ultrafinitism in math cannot be consistent, unlike the many variate form of finitism. Comp is arguably a form of finitism at the ontological level, yet an ultra-infinitism, if I can say, at the epistemological level. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Torngy, How many numbers do you think exist between 0 and 1? Certainly not only the ones we define, for then there would be a different quantity of numbers between 1 and 2, or 2 and 3. Jason On Thu, Jun 4, 2009 at 10:27 AM, Torgny Tholerus tor...@dsv.su.se wrote: Brian Tenneson skrev: Torgny Tholerus wrote: It is impossible to create a set where the successor of every element is inside the set, there must always be an element where the successor of that element is outside the set. I disagree. Can you prove this? Once again, I think the debate ultimately is about whether or not to adopt the axiom of infinity. I think everyone can agree without that axiom, you cannot build or construct an infinite set. There's nothing right or wrong with adopting any axioms. What results is either interesting or not, relevant or not. How do you handle the Russell paradox with the set of all sets that does not contain itself? Does that set contain itself or not? My answer is that that set does not contain itself, because no set can contain itself. So the set of all sets that does not contain itself, is the same as the set of all sets. And that set does not contain itself. This set is a set, but it does not contain itself. It is exactly the same with the natural numbers, BIGGEST+1 is a natural number, but it does not belong to the set of all natural numbers. The set of all sets is a set, but it does not belong to the set of all sets. What the largest number is depends on how you define natural number. One possible definition is that N contains all explicit numbers expressed by a human being, or will be expressed by a human being in the future. Amongst all those explicit numbers there will be one that is the largest. But this largest number is not an explicit number. This raises a deeper question which is this: is mathematics dependent on humanity or is mathematics independent of humanity? I wonder what would happen to that human being who finally expresses the largest number in the future. What happens to him when he wakes up the next day and considers adding one to yesterday's number? This is no problem. If he adds one to the explicit number he expressed yesterday, then this new number is an explicit number, and the number expressed yesterday was not the largest number. Both 17 and 17+1 are explicit numbers. -- Torgny Tholerus --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries 2
Hi Kim,
On 04 Jun 2009, at 14:28, kimjo...@ozemail.com.au wrote:
OK - I find this quite mind-blowing; probably because I now
understand it for the first
time in my life. So how did it get this rather ridiculous name of
square root? What's it
called in French?
Racine carrée. Literally square root.
It comes from the fact that in elementary geometry the surface or area
of a square which sides have length x, is given by x*x, also written
x^2, which is then called the square of x. Taking the square root
of a number, consists in doing the inverse of taking the square of a
number. It consists in finding the length of a square knowing its area.
Mathematician and especially logician *can* use arbitrary vocabulary.
It is the essence of the axiomatic method in pure mathematics that
what is conveying does not depend on the term which are used. Hilbert
said once that he could have use the term glass of bear instead of
line in his work in geometry.
A = {x such that x is even and smaller than 100} = {x ⎮ x is even
x
special character, abbreviating such that, and I hope it goes
through the mail.
Just an upright line? It comes through as that. I can't seem to get
this symbol happening so I will
use such that
Yes, such that is abbreviated by an upright line. Sometimes also by
a half circle followed by a little line, but I don't find it on my
palette!
If not I will use such that, or s.t., or things like that.The
expression {x ⎮ x is even} is
literally read as: the set of objects x, (or number x if we are in
a context where we talk
about numbers) such that x is even.
Exercise 1: Could you define in intension the following infinite
set C = {101, 103, 105,
...}C = ?
C = {x such that x is odd and x 101}
Correct.
Exercise 2: I will say that a natural number is a multiple of 4 if
it can be written as 4*y,
for some y. For example 0 is a multiple of 4, (0 = 4*0), but also
28, 400, 404, ... Could
you define in extension the following set D = {x ⎮ x 10 and x is
a multiple of 4}?
D = 4*x where x = 0 but also { 1, 2, 3, 4, 8 }
Hmm...
Marty made a similar error. D is a set. May be you wanted to say:
D = {4*x where x = 0 but also { 1, 2, 3, 4, 8 }}. But this does not
make much sense. Even if I try to imagine favorably some meaning, I
would say that it would mean that D is the set of numbers having the
shape 4*x (that is capable of being written as equal to 4*x for some
x), and such that x belongs to {0, 1, 2, 3, 4, 8}.
A proper way to describe that set would be
D = {y such that y = 4x and x belongs-to {0, 1, 2, 3, 4, 8}}.
But that would makes D = {0, 4, 8, 12, 32}.
The set D = {x ⎮ x 10 and x is a multiple of 4} is just, in
english, the set of natural numbers which are little than 10 and which
are a multiple of 4. The only numbers which are little than 10, and
multiple of 4 are the numbers 0, 4, and 8. D = {0, 4, 8}.
I now realise I am doomed for the next set of exercises because I
cannot get to the special
symbols required (yet). As I am adding Internet Phone to my system,
I am currently using an
ancient Mac without the correct symbol pallette while somebody
spends a few days to flip a single
switch...as soon as I can get back to my regular machine I will
complete the rest.
Take it easy. No problem.
In the meantime I am enjoying the N+1 disagreement - how refreshing
it is to see that classical
mathematics remains somewhat controversial!
The term is a bit too strong. It is a bit like if I told you that I
am Napoleon, and you conclude that the question of the death of
Napoleon is still controversial. I exaggerate a little bit to make my
point, but I know only two ultrafinitists *in math*, and I have never
understood what they mean by number, nor did I ever met someone
understanding them.
What makes just a little bit more sense (and I guess that's what
Torgny really is) is being ultrafinitist *in physics*, and being
physicalist. You postulate there is a physical universe, made of a
finite number of particles, occupying a finite volume in space-time,
etc. Everything is finite, including the everything.
Then by using the unintelligible identity thesis (and thus
reintroducing the mind-body problem), you can prevent the comp white
rabbits inflation. Like all form of materialism, this leads to
eliminating the person soon or later (by the unsolvability of the mind-
body problem by finite means). Ultrafinitist physicalism eliminates
also mathematics and all immaterial notions, including all universal
machines. Brrr...
The real question is do *you* think that there is a biggest natural
number? Just tell me at once, because if you really believe that
there is a biggest natural number, I have no more clues at all how you
could believe in any of computer science nor UDA.
Remember that Thorgny pretends also to be a zombie. It has already
RE: The seven step-Mathematical preliminaries
Date: Thu, 4 Jun 2009 15:23:04 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Quentin Anciaux skrev: If you are ultrafinitist then by definition the set N does not exist... (nor any infinite set countably or not). All sets are finite. It it (logically) impossible to construct an infinite set. What do you mean by construct? Do we have to actually write out or otherwise physically embody every element? Why can't we think of a particular set as just a type of rule that, given any possible element, tells you whether or not that element is a member or not? In this case there's no reason the rule couldn't be such that there are an infinite number of possible inputs that the rule would identify as valid members. You can construct the set N of all natural numbers. But that set must be finite. What the set N contains depends on how you have defined natural number. How do *you* define natural number, if not according to the usual recursive rule that 1 is a natural number and that if N is a natural number, N+1 is also a natural number? Hopefully you agree that there can be no finite upper limit on possible inputs you could give this rule that the rule would identify as valid natural numbers? I think your claim would be that simply describing the rule is not a valid way of constructing the set of natural numbers. If so, why *isn't* it valid? *You* may prefer to adopt the rule that we should only be allowed to call something a set if we can actually write out every member, but do you have any argument as to why it's invalid for the rest of us to define sets simply as general rules that decide whether a given input is a member or not? This seems more like an aesthetic preference on your part rather than something you have a compelling philosophical argument for (or at least if you have such an argument you haven't provided it). Jesse --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
