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 12:00 On Jul 1, 9:31 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > On Jun 29, 5:28 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > >> On Jun 29, 4:23 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > >> > I'm not skeptical (in the way that Charlie-Boo is skeptical) about > >> > the provability of Con(PA) in either ZFC or PRA+epsilon_0. But I > >> > am suspicious of the fact that the induction needs only to be > >> > taken up to epsilon_0, > > >> As to ZFC, you don't need such fancy stuff as epsilon_0. Just do the > >> routine proof that with the system of omega with 0, successor, > >> addition, and multiplication we get a model of all the PA axioms. > > > The problem isn't to conclude that a model exists, using ZFC. The > > problem is to prove that PA is consistent, using ZFC. > > If you admit that ZF can express "structure S is a model for theory T", > then have a go at working out how the provable existence of a model > shows consistency. > > Hint: a semantic definition of consistency of T is that there is > no formula P such that P and ~P are logical consequences of T. > > Hint: recall the definition of satisfaction of a negated formula > in a structure. > > Hint: try using FOL. > > Is that enough hints? Instead of nibbling at the edges and playing games, how about a high level summary of the proof and how ZFC axioms are needed to formalize that? And then give a reference to where someone has carried it out? C-B > > C-B > > >> MoeBlee > > -- > Alan Smaill
From: MoeBlee on 1 Jul 2010 12:02 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. > 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've given you a reference. 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: 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 |