Hi Brent, I have joined you last two posts,
Le 31-août-07, à 17:55, Brent Meeker a écrit : >> Yes. I can accept that PA is a description of counting. But PA, per >> se, >> is not a description of PA. With your term: I can accept arithmetic is >> a description of counting (and adding and multiplying), but I don't >> accept that arithmetic (PA) is a description of arithmetic (PA). Only >> partially so, and then this is not trivial at all to show (Godel did >> that). > > Right. PA is description of arithmetic and Godel showed that part of > PA could be described within PA. OK. And then PA can reason on PA (without any new axioms). >>> Do you see why I think your objection was a non-sequitur? >> >> >> But then how do you distinguish arithmetic and Arithmetic? How to you >> distinguish a description of counting (like PA or ZF) and a >> description >> of a description of counting, like when Godel represents arithmetic in >> arithmetic, or principia mathematica *in* principia mathematica ... > > Yes, that's the question. If arithmetic is all that is provable from > PA it is well defined. Sure. > But Arithmetic isn't well defined. Why? It is just the set of arithmetical sentences which are satisfied in the structure (N, +, *). That set is not recursively enumerable (mechanically enumerable), but non RE sets abound in math. Do you accept that classical logic works on number? If yes, it is simple to define Arithmetic in naive set theory, as you can define it in formal set theory (ZF). Recall that arithmetical truth (Arithmetic) is even just a tiny part of mathematical truth. Analytical truth, second-order logic truth, etc... are vastly bigger. Most usual categories are still bigger, ... > It's lots of different sets of propositions that are provable from > PA+Something. Sure, and if that Something is sound, this gives sequences of approximation of Arithmetic. Arithmetic, the set of true arithmetical sentence is productive (like the set of growing functions I have described to Tom). It means that not only Arithmetic is not recursively enumerable, but it is constructively so! For each RE set W_i which is propose as a candidate for an enumeration of Arithmetic, you can find an element in Arithmetic (a true sentence) which does not belong to W_i. This is a version of incompleteness. > If that Something is a Godel numbering scheme then there is a mapping > between proofs and arithmetic propositions. I don't understand. The goal of a numbering scheme is to study what PA can say about PA, without adding any "something" to PA. > >> >> It is different. In Peano Arithmetic, a number like 7 is usually >> represented by an expression like sssssss0 (or >> s(s(s(s(s(s(s(s(0))))))). >> But in arithmetical meta-arithmetic, although the number 7 is still >> represented by sssssss0, the representing object "sssssss0" will be >> itself represented by the godel number of the expression "sssssss0", >> which will be usually a huge number like >> (2^4)*(3^4)*(5^4)*(7^4)*(11^4)*(13^4)*(17^4)*(19^5), where 4 is the >> godel number of the symbol "s" and 5 is the godel number of the symbol >> "0". Prime numbers are used so that by Euclid's fundamental theorem,I >> mean the uniqueness of prime decomposition of numbers (Euclid did not >> get it completely I know) we can associate a unique number to finite >> strings of symbols. >> >> In PA the symbols "0", "+" etc. make it possible to describe numbers >> and counting. Meta-PA is a theory in which you have to describe proofs >> and reasoning. Then it is not entirely obvious that a big part of >> Meta-PA can be described in the language of PA. > > But this doesn't make it the same thing as PA. It is functionnally isomorphe. Like a comp doppelganger. It is a third person self. PA cannot prove that PA is PA, but can bet on it correctly by chance. Then you can mathematically study what PA can prove on PA, or what a (correct) machine can prove about herself, from some correct third person description of herself. Don't confuse that third person self with the first person whose "self" has not even a name. > >> This makes possible for >> PA to reason about its own reasoning abilities, and indeed to discover >> that "IF there is no number describing a proof of a falsity THEN I >> cannot prove that fact", for example. This shows that wonderful thing >> which is that PA can, by betting interrogatively on its own >> consistency, infer its own limitation with respect to the eventually >> never completely and effectively describable Arithmetic (arithmetical >> truth). > > But it is this incompleteness and indescribability of Arithmetic which > causes me to think that it doesn't exist. "arithmetic" (PA) is incomplete (provably so by us, as far as we are sound lobian machine), but it is describable, already by a simpler than us machine like ZF. Then ZF cannot define a notion of set theoretical truth. It is general: no sound machine M can ever describe or define his own global notion of M-truth. You can't infer from this that such notion are senseless. All notion of M-truth can be define by some sound lobian machine M' with M' far more complex than M. > You are just betting on it. I look at it this way: > > objects = things we observe to exist. 'course a platonist stops here. He/she would say appearance = things we observe. I don't thing we ever see "existence" of something. That is always an inference, *hopefully* stable. (unlike in night dreams for example). > counting = a physical process associating objects I can agree that when a human counts, there is some physicalness involved. But counting has more general mathematical grounding. That is what PA is about. > arithmetic = propositions about counting Yes, but Arithmetic too. > Peano's axioms = a description of arithmetic Nooooooo .... Just a little machine trying to scratch a tiny part of it. (I guess you mean Arithmetic), because arithmetic is just the theorems of PA: most of the time I identify a theory, with its set of theorems. Like I say <>t is in G* but not in G .... > PA+Godel = a description of proofs in PA > meta-mathematics = description of PA+Godel and other math. ? Metamathematics makes sense only because it can be arithmetized. Even the metamathematics of set theory. > . > . > meta^N-mathematics = description of meta^(N-1)-mathematics You can make the whole tower already in PA, or in richer theories if you want more theorems, but then you climb just on the ignorance space of mathematics. The interest of the arithmetization of arithmetic is that you collapse the level with the meta-level. > > So, do I need Arithmetic? Arithmetic is like qualia or pain. Perhaps we don't need that, but we just cannot put that under the rug. Almost all of math would disappear without Arithmetic. >> Let me sum up by singling out three things which we should not be >> confused: >> > > OK. I don't think I'm confused about them. > >> >> 1) A theory about numbers/machines, like PA, ZF or any lobian machine. >> (= finite object, or mechanically enumerable objet) >> >> 2) Arithmetical truth (including truth about machine). (infinite and >> complex non mechanically enumerable object) > > Is this the set of all (countably infinite) true propositions about > the natural numbers? Is the existence of this set a matter of faith? For a machine like PA, who happens to talk only on numbers, yes, it is a matter of faith for him/she/it. But for a lobian machine like ZF, the esistence of Arithmetic is a matter of proof. >> Only a meta-theory *about* PA, can distinguish PA and arithmetical >> truth. But then Godel showed that sometimes a meta-theory can be >> translated in or by the theory/machine. Rich theories/machine have >> indeed self-referential abilities, making it possible for them to >> guess >> their limitations. > > Are there not infinitely many Godel numbering schemes. Those defining PA in PA will be equivalent, unless they describe intensionnal variants like the hypostases, but then that is really new non equivalent points of view. Usually it is said that the choice of the numbering does not matter. > >> By doing so, such machines infer the existence of >> something transcendenting <transcending> (if I can say) themselves. >> >> OK? > > OK. All right, Bruno 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 [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
