How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Charlie-Boo on 1 Jul 2010 12:07 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. > > 1. Reference doesn't have it. > > 2. No reference. > > 3. Talks about a proof of something else in ZFC. > > 4. Isn't carried out in ZFC. > > I just gave you the reference to Hinman's book. (It won't do you any > good, though, since even if you got it, you wouldn't go through the > steps in the book leading up to said proof; also because the book is > at a somewhat advanced level that would be difficult for someone, such > as you, who has not first established an understanding of certain > basics in symbolic logic.) Didn't you read my response? Hinman doesn't refer to ZFC's axioms at all in his proof. He even admits that. Did YOU read it? C-B > MoeBlee
From: Frederick Williams on 1 Jul 2010 12:16 Charlie-Boo wrote: > > The next thing to actually do is to show how or why PA can't prove PA > consistent in detail Hilbert & Bernays do that. -- I can't go on, I'll go on.
From: Charlie-Boo on 1 Jul 2010 12:17 On Jul 1, 12:02 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jul 1, 12:31 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > On Jun 30, 5:43 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > On Jun 29, 11:18 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > > > How about telling me the title of a book or article in which PA is > > > > proved consistent using only ZFC? > > > > Peter Hinman, 'Fundamentals Of Mathematical Logic' pg. 557. Theorem > > > 6.6.9. > > > Hinman doesn't carry out any of his arguments in ZFC - he never > > mentions any of the axioms of ZFC at all - maybe I missed somthing - > > Yes, you did. Perhaps your copy is missing page 462 where Hinman lists > the ZF axioms. Oh no!!!! MoeBlee, mon ami, you know that he doesn't use any of them in his proof, right? > > do you see any references to any ZFC axioms in his proof (other than > > perhaps Peano's Axioms in the form of the axiom of infinity > > PA doesn't have an axiom of infinity or anything like it. I'm saying that ZFC has that axiom, of course. You know that it does. As far as PA not having "anything like it" goes, don't be silly. > 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!) YOU DON'T HAVE A LEGITIMATE REFERENCE. ADMIT IT! You're just like Frederick, one of the first to lie about this with his own BS reference. C-B > I don't have time to tutor you, especially > on such basic matters that you could settle for yourself were you only > willing to learn the subject properly from page 1 and with copies of > books not missing important pages. > > MoeBlee
From: Frederick Williams on 1 Jul 2010 12:23 Charlie-Boo wrote: > > Of course infinity = PA = arithmetic. Of course. PA doesn't mention infinity. Just because these chaps 0, 0', 0'', 0''', ... exist, doesn't mean that this chap {0, 0', 0'', 0''', ...} exists. Which--speaking loosely--is why ZF needs an axiom of infinity and PA doesn't. -- I can't go on, I'll go on.
From: MoeBlee on 1 Jul 2010 12:25
On Jul 1, 9:01 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jun 29, 10:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Aatu Koskensilta wrote: > >>> Frederick Williams <frederick.willia...(a)tesco.net> writes: > >>>> Yes, you can: take Gentzen's proof (or Ackermann's etc) and formalize > >>>> it in ZFC. > >>> This is a pretty silly way of proving the consistency of PA in set > >>> theory. > >>> That PA is consistent is a triviality. > >> In what formal system is this triviality in? > > > It's a theory of Z-R, for example. Whether it's "trivial" to prove in > > Z-R depends on what strikes one as trivial. > > >> (Iow, you didn't mean > >> it's a fact that PA is syntactically consistent, did you?) > > > Consistent IS syntactically consistent. > > But there's also such thing as relative consistency proof! Yes, of course. I don't know why you're excited about that fact though. > For example, > from T = {Ax[xex] /\ ~Ax[xex]}, it's a triviality to prove the consistency > of PA, Sure, as long as there is some sentence in the language of T that we read as "PA is not consistent". Of course, such a proof does not in itself give evidence that there is a PA proof of a formula P&~P. Rather, such a proof gives evidence merely that in T there is a certain derivation of a formula that we are reading as "PA is not consistent". > but should I proclaim that PA is consistent, as in, "that PA is > consistent is a triviality", as Aatu put it? Right, we agree you should not take such a proof as evidentiary in that way. But, just to be clear (since I'm not sure exactly what you're saying) Aatu is not claiming that you should. > The question I had for him was a clarification request to see if he meant > PA is really consistent, Yes, he means that PA is consistent, really consistent. > or if he meant that was just a relative consistency > proof he had referred to. The above you referred to is not a relative consistency. A relative consistency is of the form: T |- G consistent -> G* consistent The proof you mentioned is of the form: T |- G consistent. Anyway, Aatu is not saying just that there exists a relative consistency proof nor just that, say, ZF or some other formal system proves Con(PA), but rather he's saying that PA IS consistent. He's saying that aside from whatever FORMAL proofs, PA is consistent - PERIOD. His basis is for that is not a FORMAL proof, but rather his conviction that the axioms of PA are true (and not even in confined to a FORMAL model theoretic sense of truth, but rather that the axioms are simply true about the natural numbers, as we (editorial 'we') understand the natural numbers even aside from any formalization. Haven't you read Franzen's incompleteness book? > (You should read people's conversation more carefully, before jumping to > conclusion whether or not people understand this or that.) I didn't post anything that shows lack of context of the conversation. > > Here's one among equivalent definitions: > > > DEFINITION OF CONSISTENT: > > > A set of formulas S is in a language is consistent iff there is no > > formula P and the negation of P in S. Typo: delete the first 'is'. > > PERIOD. > > I was about to ignore your incorrectness here, but the tallness of your > ending "PERIOD" seemed defying any, say, "forgiveness". So here it is. Nope, I'm not gonna go down the suckhole with you again. MoeBlee |