How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
in [Math]
|
Prev: ? theoretically solved
Next: How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
From: MoeBlee on 30 Jun 2010 11:15 On Jun 29, 11:45 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Tue, 29 Jun 2010 08:02:06 -0700 (PDT), MoeBlee <jazzm...(a)hotmail.com> said: > > 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. Would you recommend a book (or site) where I can read it in > > all details? > > Hm, don't really know of any books. There's a recent article by Kaye > and Wang called "On Interpretations of Arithmetic and Set Theory" that > occurred in 2007 or 2008 in the Notre Dame Journal of Formal Logic that > discusses the mapping in detail. Great, thanks, I think that NDJFL is free online. I'll look. Meanwhile, I found a little bit also in Kunen's 'Foundations Of Mathematics'. MoeBlee
From: MoeBlee on 30 Jun 2010 11:17 On Jun 30, 2:44 am, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > The notion of provability in a theory is not formalizable in NAFL > theories. It must remain as a metamathematical notion. Meanwhile, metamathematical notions, including 'provability' are formalizable in such theories as Z set theory. You understand that, right? MoeBlee
From: Frederick Williams on 30 Jun 2010 11:22 MoeBlee wrote: > Great, thanks, I think that NDJFL is free online. I'll look. I don't think so, but you'll find it at Kaye's site. -- I can't go on, I'll go on.
From: MoeBlee on 30 Jun 2010 11:28 On Jun 30, 9:42 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > Hmmm. I am not very conversant with classical model theory. So > > according to you, there is a "standard model for the LANGUAGE of PA" > > even if the theory PA is inconsistent. > > No just according to me. Oopsie doopsie. I meant: NoT just according to me. As I said, if Z is inconsistent, then Z proves every formula in the language. So, if we find something to be a theorem of Z (such as the existence and uniqueness theorems that provide for a definition of the constant nicknamed "the standard model for the language of PA"), then if Z is inconsistent, PERFORCE Z proves those theorems. MoeBlee
From: Frederick Williams on 30 Jun 2010 11:43
"R. Srinivasan" wrote: > Hmmm. I am not very conversant with classical model theory. So > according to you, there is a "standard model for the LANGUAGE of PA" > even if the theory PA is inconsistent. May I infer that you have used > infinite sets to define this model? No sets, and a fortiori no infinite sets, are required (not for first-order PA anyway): the individuals are 0, 1, 2, 3, ...; the constant 0 is 0; successor is x |-> x + 1; sum is (x, y) |-> x + y; and product is (x, y) |-> x * y. Mathematicians knew all about those before model theory was invented and before Peano was an eye in his father's twinkle. > How can you do that if the theory > PA is inconsistent (which would make ZFC inconsistent as well)? The above is a model of PA whether or not it and ZFC are consistent. -- I can't go on, I'll go on. |