Re: Infinities, cardinality, diagonalisation
Thanks. I guess I agree with the your quale and betting descriptions. Finiteness is an existence statement, there exists an end. If we are talking about an actual particular group of things, we need an observer to say where the end is, thus declaring it to be finite. But it is another thing to make an abstraction of (all instances of) that occurrence (i.e. declaring an end) and call that the *concept* of fin-iteness. It is an abstraction of qualia, like a lot of math is. An aside: Some might argue that at that point you can get rid of the observer (unless it was me!), and then... you didn't need an observer in the first place... This thread is somewhat interesting, but it can get circular pretty fast. (For instance, I was going to say | has no end by definition.) I guess that's the point. But I think the bottom line is that we agree that finiteness is at the qualia level, but I would say that we can abstract it. Perhaps it is like Church Thesis? Tom Bruno Marchal wrote: Le 14-juil.-06, à 18:52, Tom Caylor a écrit : Here is where I believe the crux is: ... means you can continue to add the I as many times as you want. Actually, this is equivalent to: ... means you can continue to add the I as many times as you want and you can. It's just a little redundant to say it that way. Now A and B *know*, as well as anyone can even know, what finite means. True, but unprovable. With comp you are betting here. All they have to do is perform some experimentation to get the idea that, after a while of adding I they eventually get tired and/or loose interest, so they have to *stop*. Yes but my friend B, which is an angel, a cousin of the analytical second order arithmetic with the omega rule. He is tired after counting up to number like |||...|||...|||... .... What's so difficult about understanding what stopping is? I am not denying we have some intuition of that. Just pointing that mathematicians can show we cannot define what finite means through first order logic, and then second order logic builds on that intuition, so that really finite is not a notion we can define in any finite way. Nor can we define NOT finite, that is what infinite means. Even the word finite has fin in it, i.e. end. The notion is defined by invariance. Relative one. You can imagine something stopping compare to something which does not stop. Something similar (invariant) is happening (adding I at one step is considered the same action as adding I in another step) Actually adding | at the end of . giving |||| is different from adding | at the end of |||. and then the invariance disappears, i.e. the adding of the I is no longer happening. Yes but when? I know you and me know that. The point is that we cannot explain it without admitting at the start that we know that. that has the type of a quale. 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Infinities, cardinality, diagonalisation
Le 14-juil.-06, à 18:52, Tom Caylor a écrit : Here is where I believe the crux is: ... means you can continue to add the I as many times as you want. Actually, this is equivalent to: ... means you can continue to add the I as many times as you want and you can. It's just a little redundant to say it that way. Now A and B *know*, as well as anyone can even know, what finite means. True, but unprovable. With comp you are betting here. All they have to do is perform some experimentation to get the idea that, after a while of adding I they eventually get tired and/or loose interest, so they have to *stop*. Yes but my friend B, which is an angel, a cousin of the analytical second order arithmetic with the omega rule. He is tired after counting up to number like |||...|||...|||... .... What's so difficult about understanding what stopping is? I am not denying we have some intuition of that. Just pointing that mathematicians can show we cannot define what finite means through first order logic, and then second order logic builds on that intuition, so that really finite is not a notion we can define in any finite way. Nor can we define NOT finite, that is what infinite means. Even the word finite has fin in it, i.e. end. The notion is defined by invariance. Relative one. You can imagine something stopping compare to something which does not stop. Something similar (invariant) is happening (adding I at one step is considered the same action as adding I in another step) Actually adding | at the end of . giving |||| is different from adding | at the end of |||. and then the invariance disappears, i.e. the adding of the I is no longer happening. Yes but when? I know you and me know that. The point is that we cannot explain it without admitting at the start that we know that. that has the type of a quale. 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Infinities, cardinality, diagonalisation
Hi Quentin, Tom and List,
Of course, N is the set of finite positive integers:
N = {0, 1, 2, 3, ...}.
An infinite set A is countable or enumerable if there is a computable bijection between A and N.
Forgetting temporarily the number zero, all finite number can be put in the shapes:
|
||
|||
|
||
|||
...
This raises already an infinitely difficult problem: how to define those finite numbers to someone who does not already have some intuition about them. The answer is: IMPOSSIBLE. This is part of the failure of logicism, the doctrine that math can be reduced to logic. technically this can be explained through mathematical logic either invoking the incompleteness phenomenon, or some result in model theory (for example Lowenheim-Skolem results).
But it is possible to experience somehow that impossibility by oneself without technics by trying to define those finite sequence of strokes without invoking the notion of finiteness.
Imagine that you have to explain the notion of positive integer, or natural number greater than zero to some extraterrestrials A and B. A is very stubborn, and B is already too clever (as you will see).
So, when you present the sequence:
| || ||| ...
A replies that he has understood. Numbers are the object |,and the object ||, and the object |||, and the object ...”. So A conclude there are four numbers. You try to correct A by saying that ... does not represent a number, but does represent some possibility of getting other numbers by adding a stroke | at their end. From this A concludes again that there is four numbers: |, ||, |||, and . You try to explain A that ... really mean to can continue to add the |; so A concludes that there are five numbers now. So you will try to explain to A that ... means you can continue to add the I as many times as you want. From this A will understand that the set of numbers is indefinite: it is
{|, ||, |||, , |} or {|, ||, |||, , |, ïï} or some huge one but similarly ... finite.
Apparently A just doesn't grasp the idea of the infinite.
B is more clever. After some time he seems to grasp the idea of ..., and apparently he does understand the set {|, ||, |||, , |, ...}. But then, like in Tom's post, having accepted the very idea of
infinity through the use of ..., B, in some exercise, can accept the infinite object
|...
itself as a number. How will you explain him that he has not the right to take this as a finite number. He should add that the rule, consisting in adding a stroke | at the end of a number (like |||), can only be applied a finite number of times. OK, but the problem was just that: how to define a finite number
The modern answer is that this is just impossible. The set of positive integers N cannot be defined univocally in any finite way.
This can take the form of some theorem in mathematical logic. For example: it is not possible to define the term finite in first order classical logic. There is not first order logic theory having finite model for each n, but no infinite models.
You can define finite in second order logic, but second order logic are defined through the intuition of finiteness/non-finiteness, so this does not solve the problem.
This can be used to show that comp will make the number some absolute mystery.
Now, note that B, somehow, can consider the generalized number:
|...
as a number. Obviously, this corresponds to our friend the *ordinal omega*. From the axiom that you get a number by adding a stroke at its end: you will get
omega+1, as
|...|
omega+2, as
|...||
omega+3
|...|||
...
omega+omega
||...||...
...
omega+omega+omega
||...||...||...
...
omega*omega
|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||...|||... ...
... and this generate a part of the constructive countable ordinals.
And we stay in the domain of the countable structure, unless you decide to go up to the least non countable ordinals and beyond. For doing this properly you need some amount of (formal) set theory. In all case, what ... expressed is unavoidably ambiguous.
Bruno
Le 13-juil.-06, à 15:29, Quentin Anciaux a écrit :
Hi, thank you for your answer.
But then I have another question, N is usually said to contains positive integer number from 0 to +infinity... but then it seems it should contains infinite length integer number... but
Re: Infinities, cardinality, diagonalisation (errata)
Le 14-juil.-06, à 14:34, Bruno Marchal a écrit :
Hi Quentin, Tom and List,
Of course, N is the set of finite positive integers:
N = {0, 1, 2, 3, ...}.
An infinite set A is countable or enumerable if there is a computable
bijection between A and N.
Please suppress the computable in that last sentence.
Bruno
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Re: Infinities, cardinality, diagonalisation
Hi, thank you for your answer.But then I have another question, N is usually said to contains positive integer number from 0 to +infinity... but then it seems it should contains infinite length integer number... but then you enter the problem I've shown, so N shouldn't contains infinite length positive integer number. But if they aren't natural number then as the set seems uncountable they are in fact real number, but real number have a decimal point no ? so how N is restricted to only finite length number (the set is also infinite) without infinite length number ? Thanks,QuentinOn 7/13/06, Tom Caylor [EMAIL PROTECTED] wrote: I think my easy answer is to say that infinite numbers are not in N.Ilike to think of it with a decimal point in front, to form a numberbetween 0 and 1.Yes you have the rational numbers which eventually have a repeating pattern (or stop).But you also have in among themthe irrational numbers which are uncountable. (Hey this reminds me ofthe fi among the Fi.)To ask what is the next number after an infinite number, like 1...1... is similar asking what is the next real number after0.1...1...Tom --~--~-~--~~~---~--~~ 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Infinities, cardinality, diagonalisation
Quentin Anciaux wrote: Hi, thank you for your answer. But then I have another question, N is usually said to contains positive integer number from 0 to +infinity... but then it seems it should contains infinite length integer number... but then you enter the problem I've shown, so N shouldn't contains infinite length positive integer number. But if they aren't natural number then as the set seems uncountable they are in fact real number, but real number have a decimal point no ? so how N is restricted to only finite length number (the set is also infinite) without infinite length number ? Thanks, Quentin The ordinary definitions of the natural numbers or the real numbers do not include infinite numbers, but in at least some versions of nonstandard analysis (which as I understand it is basically a way of allowing 'infinitesimal' quantities like the dx in dx/dy to be treated as genuine numbers) you can have such infinite numbers (I believe they're the reciprocal of infinitesimals). I know the system of the hyperreals contains them, see http://mathforum.org/dr.math/faq/analysis_hyperreals.html for some more info. I'm not sure if infinite hyperreal numbers have the sort of decimal expansion that you suggest though, skimming that page it seems that infinite hyperreals are identified with the limits of sequences that sum to infinity, like 1+2+3+4+..., but different sequences can sometimes correspond to the same hyperreal number, you need some complicated set theory analysis to decide which series are equivalent. Since the hyperreals contain all the reals, the set must be uncountable...I don't know if it would be possible to just consider the set of infinite hyperreal integers or not, and if so whether this set would have the same cardinality as the continuum. 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Infinities, cardinality, diagonalisation
N is defined as the positive integers, {0, 1, 2, 3, ...}, i.e. the
*countable* integers. (I am used to starting with 1 in number
theory.) N does not include infinity, neither the countable infinity
aleph_0 nor any other higher infinity. Infinite length integers
fall into this category of infinities. As you have shown, the infinite
length integers cannot be put in a one-to-one correspondence with N.
This is the definition of uncountable. However, just because the set
of infinite length integers is uncountable, or even equivalent in
cardinality to the set of real numbers, doesn't mean they are real
numbers. There are other sets that have the same cardinality as the
set of real numbers, 2^aleph_0, for instance the set of all subsets of
N. Granted, there are (undecidable) mysteries involved, as Jesse has
alluded to, when we start trying to sort out all of the possible
infinite beasts, and this is partly why the Continuum Hypothesis is
such a mystery. But with the given definitions of countable and
uncountable, infinite length integers are uncountable, and so not in
N. Conversely, just because you can *start* counting the reals
(starting with the rationals), and you can *start* counting the
infinite integers, and it would take forever (just like counting
the integers would take forever) doesn't mean they are countable. We
need some kind of definition like the one-to-one correspondence
definition of Cantor to distinguish countable/uncountable.
Tom
Quentin Anciaux wrote:
Hi, thank you for your answer.
But then I have another question, N is usually said to contains positive
integer number from 0 to +infinity... but then it seems it should contains
infinite length integer number... but then you enter the problem I've shown,
so N shouldn't contains infinite length positive integer number. But if they
aren't natural number then as the set seems uncountable they are in fact
real number, but real number have a decimal point no ? so how N is
restricted to only finite length number (the set is also infinite) without
infinite length number ?
Thanks,
Quentin
On 7/13/06, Tom Caylor [EMAIL PROTECTED] wrote:
I think my easy answer is to say that infinite numbers are not in N. I
like to think of it with a decimal point in front, to form a number
between 0 and 1. Yes you have the rational numbers which eventually
have a repeating pattern (or stop). But you also have in among them
the irrational numbers which are uncountable. (Hey this reminds me of
the fi among the Fi.)
To ask what is the next number after an infinite number, like
1...1... is similar asking what is the next real number after
0.1...1...
Tom
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Re: Infinities, cardinality, diagonalisation
Technically, I should say that countable means that the set can be put
into a one-to-one correspondence with *a subset of* N, to include
finite sets.
Tom
Tom Caylor wrote:
N is defined as the positive integers, {0, 1, 2, 3, ...}, i.e. the
*countable* integers. (I am used to starting with 1 in number
theory.) N does not include infinity, neither the countable infinity
aleph_0 nor any other higher infinity. Infinite length integers
fall into this category of infinities. As you have shown, the infinite
length integers cannot be put in a one-to-one correspondence with N.
This is the definition of uncountable. However, just because the set
of infinite length integers is uncountable, or even equivalent in
cardinality to the set of real numbers, doesn't mean they are real
numbers. There are other sets that have the same cardinality as the
set of real numbers, 2^aleph_0, for instance the set of all subsets of
N. Granted, there are (undecidable) mysteries involved, as Jesse has
alluded to, when we start trying to sort out all of the possible
infinite beasts, and this is partly why the Continuum Hypothesis is
such a mystery. But with the given definitions of countable and
uncountable, infinite length integers are uncountable, and so not in
N. Conversely, just because you can *start* counting the reals
(starting with the rationals), and you can *start* counting the
infinite integers, and it would take forever (just like counting
the integers would take forever) doesn't mean they are countable. We
need some kind of definition like the one-to-one correspondence
definition of Cantor to distinguish countable/uncountable.
Tom
Quentin Anciaux wrote:
Hi, thank you for your answer.
But then I have another question, N is usually said to contains positive
integer number from 0 to +infinity... but then it seems it should contains
infinite length integer number... but then you enter the problem I've shown,
so N shouldn't contains infinite length positive integer number. But if they
aren't natural number then as the set seems uncountable they are in fact
real number, but real number have a decimal point no ? so how N is
restricted to only finite length number (the set is also infinite) without
infinite length number ?
Thanks,
Quentin
On 7/13/06, Tom Caylor [EMAIL PROTECTED] wrote:
I think my easy answer is to say that infinite numbers are not in N. I
like to think of it with a decimal point in front, to form a number
between 0 and 1. Yes you have the rational numbers which eventually
have a repeating pattern (or stop). But you also have in among them
the irrational numbers which are uncountable. (Hey this reminds me of
the fi among the Fi.)
To ask what is the next number after an infinite number, like
1...1... is similar asking what is the next real number after
0.1...1...
Tom
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Re: Infinities, cardinality, diagonalisation
I think my easy answer is to say that infinite numbers are not in N. I like to think of it with a decimal point in front, to form a number between 0 and 1. Yes you have the rational numbers which eventually have a repeating pattern (or stop). But you also have in among them the irrational numbers which are uncountable. (Hey this reminds me of the fi among the Fi.) To ask what is the next number after an infinite number, like 1...1... is similar asking what is the next real number after 0.1...1... Tom --~--~-~--~~~---~--~~ 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 [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
