How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: MoeBlee on 1 Jul 2010 13:01 On Jul 1, 11:07 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jul 1, 11:53 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 30, 11:17 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > The problem isn't to conclude that a model exists, using ZFC. The > > > problem is to prove that PA is consistent, using ZFC. > > > I've gone over this already with you. ZFC proves that if a theory has > > a model then the theory is consistent. It's an extremely simple > > exercise I proposed you might think about and complete in about two > > minutes. > > Then there's no reason for not giving it. Bravado is no substitute > for Mathematics. The reason for giving it as an exercise is to get you STARTED THINKING. WHAT bravado? Come on, really. A theory is a set of sentences (all in a language) closed under entailment. If a theory T is inconsistent, then there is a sentence P such that both P and ~P are in T. But P is true in model M iff ~P is false in model M, and no sentence is both true and false in a given model M (by the definition-by-recursion function that maps sentences to true or (exclusive or) to false per a model). So if a theory is inconsistent, then the theory has no model (lest there be a sentence P that is both true and false in the model, which is impossible). Note: This does not preclude that there are models ('structures' if you prefer) for the LANGUAGE of an inconsistent theory. For any language, there are many models for that language. But if a theory is inconsistent, then there is no model in which all of the sentences of the theory are true. Now, I PROMISE myself. No more explanation of this for you. If you don't understand or have some question or objection about it. Then just study the matter. I suggest Enderton's book for this particular matter. > Didn't you read my response? Hinman doesn't refer to ZFC's axioms at > all in his proof. The axioms are used in the various steps leading up to the proof. That's how mathematics works. A proof of a theorem may rely on previously proven theorems. > He even admits that. What specific quote do you have in mind? He doesn't need ZFC. ZF is sufficient (less is sufficient too). One disclaimer: I've given you the Hinman reference since you've asked for a reference. I have not scrutinized his particular proof, since this is something I proved myself long before I got Hinman's book. Nevertheless, it is a reference as you asked for one, and you may elect or not to read all the steps in the book that lead to his exposition of why ZF proves PA is consistent. MoeBlee
From: R. Srinivasan on 1 Jul 2010 13:11 On Jul 1, 9:59 pm, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > "R. Srinivasan" wrote: > > > I didn't ask how many natural numbers are exceeded by some specific, > > fixed n. > > You asked "how many natural numbers are exceeded by some element of N?" > True. In the sense that An Em m > n For each n, there is some m that exceeds it. So how many such n's are there that are exceeded *within* N? Certainly more than finitely many. > > But never mind... > > > I am asking you to count the number of natural numbers that > > are not upper bounds for N. Since all natural numbers fit this > > description, the answer has to be "infinitely many". > > Agreed. > > > The assertion > > that infinitely many natural numbers have been exceeded within N ... > > There is nothing in N that exceeds infinitely many natural numbers. > True. But the assertion that "more than finitely many natural numbers are not upper bounds for N" translates precisely to "Infinitely many natural numbers are exceeded within N". This is the intuition behind the existence of nonstandard natural numbers, by the way. RS
From: MoeBlee on 1 Jul 2010 13:16 On Jul 1, 11:17 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > On Jul 1, 12:02 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > I've given you a reference. > > Yes, and as a Conservative you know the creed that to name a book is > to defeat your enemy - especially if it is very hard to obtain. (And > to make sure, refuse to quote from it!) WHAT are you talking about? I'm not a conservative. I didn't say I "defeated" you by naming a book. You asked for a reference and I gave you one. You asked for reference. I'm not a bookseller who knows what's hard or easy to obtain. I just happened to notice that Hinman is at least one book that provides a reference for this matter. And it seems you got the book quickly enough anyway. And whatever I said about quoting from it, it's understandable that I'd rather let you read for yourself since (1) It's hard to quote in ASCII all his speical symbols, plus give all the special definitions of his notation. (2) The whole page itself is not comprehensible if one has not read and understoof the steps leading up that page, some of which go back perhaps hundreds of pages previous in the book. (3) I didn't even say that I'm an expert in Hinman's own exposition. Indeed, I only gave it as a reference if you wish to study for yourself. Personally, I've studied mostly for other books, and from that study am able to prove Z |- Con(PA) myself. (4) I don't have unlimited time to tutor you in this subject. I gave you a reference; I didn't thereby also promise to help you read it. You seem to THRIVE on making these conversations as unproductive as possible. I need to stop (I'm BEGGING myself to stop). MoeBlee
From: MoeBlee on 1 Jul 2010 13:21 On Jul 1, 12:11 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > the assertion that "more than finitely many natural numbers > are not upper bounds for N" translates precisely to "Infinitely many > natural numbers are exceeded within N". Okay. > This is the intuition behind the existence of nonstandard natural > numbers, by the way. There are lots of routes to nonstandard models. I don't know how you determined the above is "the intuition". (And that is not a request that you post a bunch more of your claims about NAFL.) One simple route to non-standard models is the compactness theorem. MoeBlee
From: R. Srinivasan on 1 Jul 2010 14:03
On Jul 1, 10:21 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jul 1, 12:11 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > > > the assertion that "more than finitely many natural numbers > > are not upper bounds for N" translates precisely to "Infinitely many > > natural numbers are exceeded within N". > > Okay. > > > This is the intuition behind the existence of nonstandard natural > > numbers, by the way. > > There are lots of routes to nonstandard models. I don't know how you > determined the above is "the intuition". (And that is not a request > that you post a bunch more of your claims about NAFL.) > > One simple route to non-standard models is the compactness theorem. > Of course. The route I mentioned is via Edward Nelson's Principle of Idealization in his Internal Set Theory. RS |