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 30 Jun 2010 10:46 On Jun 28, 11:18 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Mon, 28 Jun 2010 15:46:25 -0700 (PDT), MoeBlee <jazzm...(a)hotmail.com> > said: > > > One thing I don't know how to do is show the mutual-interpretability > > of PA and Y=ZF-"ax inf"+"~ax inf" > > > One direction seems not too difficult: interpreting PA in Y. > > > But how do we interpret Y in PA? Specifically, how do we define 'e' in > > PA and then prove, in PA, all the axioms of Y as interpreted in the > > language of PA? > > The best known approach uses a mapping that Ackermann defined from the > hereditarily finite sets into N that takes the empty set to 0 and, > recursively, {s_1,...s_i} to 2^(n_1) + ... + 2^(n_i), where n_i codes > s_i. For numbers n and m, let nEm iff the quotient of m/2^n is odd. > The relation E is obviously definable in PA. Ackermann showed that, by > defining the membership predicate as E, the axioms of Y are all theorems > of PA. Thanks. And if anyone is interested, it's mentioned in Kunen's 'Foundations Of Mathematics' too, but not with the full proof. Hopefully, one I'll work through it. MoeBlee
From: MoeBlee on 30 Jun 2010 10:55 On Jun 29, 8:17 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 29, 2:05 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 29, 3:44 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > If MoeBlee is > > > going to insist that Srinivasan prove that D is actually a set, then > > > maybe MoeBlee should do the same for Goedel's V and L. > > You pontificate out of IGNORANCE. Godel worked in NBG where we prove > > that there do exist proper classes. > > Also, even if in ZF, we refer to V and L as "figures of speech" that > > must resolve back to actual formulas in the language of ZF. > > In that case, if Srinivasan were to work in NBG-Infinity instead of > ZF-Infinity, would he then be allowed to talk about his "D"? Of course. And he can talk about it using a Z set theory too, as long as it is understood that it is "figure of speech", and with care so that fallacious inferences are not drawn. Anyway, I showed him how to state is axiom without referrring to a proper class, just to be clear (and I think his axiom is equivalent in context to "all sets are hereditarily finite"?). > I > see no reason that D wouldn't be a class in NBG-Infinity As long as it is written along these lines: D = {x | x is a set & ~En x in P_n(0)}. But I don't know how you would derive (I suspect you cannot): EyAx(x in y <-> ~En x in P_n(0)). Suppose you could. Then V would be in y. Contradiction. Right? MoeBlee
From: MoeBlee on 30 Jun 2010 11:03 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. 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. PERIOD. That a set of FIRST order formulas is consistent iff that set of sentences is satisfiable is a RESULT we prove. And, of course, Aatu is claiming that PA is consistent. He's been saying it for at least about a decade. What don't you understand about that? MoeBlee
From: MoeBlee on 30 Jun 2010 11:05 On Jun 29, 11:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > R. Srinivasan wrote: > > On Jun 30, 1:36 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > >> On Jun 29, 12:28 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote > > May I infer that you have used > > infinite sets to define this model? How can you do that if the theory > > PA is inconsistent (which would make ZFC inconsistent as well)? > > The answer imho is simple: they, the "standard theorists" (and I use > the phrase in a respectful way), would assert they somehow "know" > the natural numbers and this "standard model for the LANGUAGE of PA" > is just the natural numbers, collectively! No, I use no such argument. MoeBlee
From: MoeBlee on 30 Jun 2010 11:13
On Jun 29, 11:35 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > the cancer of infinity But not to fear, chief oncologist, Dr. Srinivasan has just the right chemo...but watch for side effects worse than the disease. > is so deeply ingrained in the > thinking of classical logicians that they are incapable of > appreciating any attempt to remove this cancer from logic. You're unfamiliar with even some of the most famous literature in the subject. MoeBlee |