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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