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 11:51 On Jul 1, 2:34 am, Ki Song <kiwisqu...(a)gmail.com> wrote: > Perhaps an analogy is in order! > > I feel like what Charlie-Boo is asking people to do is analogous to > asking someone to perform the addition: > > Sqrt{2}+Sqrt{3}, with a decimal precision of 10^(10^(10^100)) place. I thought I was asking for a reference to substantiate the claim that ZFC can prove PA consistent. By my reasoning, it can't. I gave my reasoning but other don't give their's for the opposing view. The next thing to actually do is to show how or why PA can't prove PA consistent in detail - take Godel's proof and "move the subroutines in- line" - see exactly what it says about the wffs, axioms, rules, theorems etc. of PA. Then apply that to ZFC. C-B > On Jul 1, 2:04 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > > Charlie-Boo <shymath...(a)gmail.com> writes: > > > To see it means to have actually created it, and its actual creation > > > would answer the very interesting question of whether ZFC can prove > > > that PA is consistent even though PA can't. > > > We already know the answer. The axioms used in the proof are a > > restricted form of comprehension, infinity, union and pairing. Spelling > > out the formalization of "PA is consistent" in the language of set > > theory would be a tedious, trivial and pointless undertaking, of no > > apparent mathematical interest whatsoever. > > > > It would substantiate your assertions. Is that of value? > > > Well, no. I can't think of any pressing reason I should want to convince > > you of anything. > > > -- > > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus- Hide quoted text - > > - Show quoted text -
From: MoeBlee on 1 Jul 2010 11:53 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. > 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.) MoeBlee
From: Charlie-Boo on 1 Jul 2010 11:53 On Jul 1, 2:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Ki Song <kiwisqu...(a)gmail.com> writes: > > Perhaps an analogy is in order! > > > I feel like what Charlie-Boo is asking people to do is analogous to > > asking someone to perform the addition: > > > Sqrt{2}+Sqrt{3}, with a decimal precision of 10^(10^(10^100)) place. > > The analogy is apt, but it's not really quite that bad. Based on nothing > at all I'd estimate that a formalization of "PA is consistent", fully > spelled-out in the language of set theory, fits in less than ten > pages. Ergo don't give as much as a hint of any detail as to how it would be constructed or any other aspect of it, aside from being certain that it can be done (and it is too complex to even think about)? C-B > But perhaps you were thinking of Bourbaki, whose term for the > number 1 already consists of 4,523,659,424,929 symbols and > 1,179,618,517,981 disambiguatory links? > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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 |