How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Frederick Williams on 27 Jun 2010 14:25 Charlie-Boo wrote: > > On Jun 27, 1:29 pm, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > Charlie-Boo wrote: > > > Hey Frederick, I bet you $449.94 that Gentzen's book doesn't contain a > > > proof that PA is consistent, carried out in ZFC. You on? > > > > No, but I do claim that it could be formalized in ZFC. > > > > > Or do you say things that you don't believe? > > > > Yes, but I don't think that's relevant here. > > How about when you said that Gentzen proved PA consistent using ZFC? The proof could be formalized in ZFC. I do not claim that Gentzen used nothing but the *language* of ZFC. -- I can't go on, I'll go on.
From: Frederick Williams on 27 Jun 2010 14:37 Charlie-Boo wrote: > For the record, ZFC/PA is often described as �ZFC Arithmetic Proofs/ > Peano Arithmetic� and abbreviated ZAPPA. Google it and you�ll see > it�s very popular. Frank-ly, my dear, I don't give a damn -- I can't go on, I'll go on.
From: apoorv on 27 Jun 2010 14:42 On Jun 27, 3:50 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > On Jun 27, 7:51 am, Tim Little <t...(a)little-possums.net> wrote:> On 2010-06-26, R. Srinivasan <sradh...(a)in.ibm.com> wrote: > > > > The theory ZF-Inf+~Inf clearly proves ~Inf ("Infinite sets do not > > > exist"). > > > Actually ~Inf does not assert "Infinite sets do not exist". It only > > asserts "there does not exist a successor-closed set containing the > > empty set". It may turn out to prove an equivalent statement under > > the rest of the axioms, but ~Inf does not actually mean "infinite sets > > do not exist". > > Please see my reply to Transfer Principle. We may sidestep this issue > by replacing ZF-Inf+~Inf in my post with a theory F which will only > admit models with hereditarily finite sets. > > > > This proof obviously implies that "There does not exist a model for > > > PA", for a model of PA must have an infinite set as its universe > > > Even if your intepretation of ~Inf were correct, all it would prove is > > that ZF-Inf+~Inf does not model PA. > > Sure. But the point is that ZF-Inf+~Inf is *the* chosen metatheory > (or model theory) of PA, in which models of PA, if they exist, can be > constructed. I happen to have chosen a metatheory which will not admit > any models of PA. In such a metatheory, PA is provably inconsistent > because we have a proof that models of PA cannot exist. > > > > Now I am sure a lot of people are going to jump up and down and > > > protest at this interpretation. But it is logical. > > > No, it is not. It exhibits a fairly elementary failure of logic. The > > statement "X models PA" implies "PA has a model". However, "X does > > not model PA" does *not* imply "PA has no model". > > What you are effectively saying is that ZF-Inf+~Inf is the "wrong" > metatheory because there *are* models of PA "out there" , meaning > outside of our chosen model theory. This is just an assertion of > Platonic existence. There is no particular reason for preferring ZF to > ZF-Inf+~Inf as a model theory for PA. The fact that the latter theory > yields an unpleasant result does not make it "wrong". > > RS Very well put. If we take it that PA captures the intuitive notion of numbers, then maybe, we should be able to interpret into PA that portion of any theory that purports to represent numbers . So, if we take the language of set theory, and the axioms Ey Ax ~x e y and Ax Ez Ay (y e z <->y e x or y=x), we have an interpretation into PA with e corresponding to <. If we want to preserve this interpretation, the axiom of infinity cannot be added to these two axioms, as it leads to a 'number' w other than 0 without a predecessor and PA does not admit of such a number.So, we have a choice between giving up infinity and and accepting PA as a 'gold standard' or accepting that induction extends across infinity to non-standard naturals. -apoorv
From: Charlie-Boo on 27 Jun 2010 14:44 On Jun 27, 1:59 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 24, 2:01 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > And the ZFC part is the DATA STRUCTURES of this "programming > > language". When programmers need to go beyond aleph-1 integers > > DAMN, you're stupid. > PROGRAMMERS NEVER need to go beyond aleph-1 integers. That's not true. Many computers print bills and they deal with decimal numbers - the number of dollars isn't always a whole number. > Indeed, they never even MAKE IT UP to aleph-ZERO integers! > COMPUTERS ARE FINITE! And so are you. > Jeezus. Good decision. C-B
From: Charlie-Boo on 27 Jun 2010 14:44
On Jun 27, 2:00 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 24, 5:01 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > Read the first 3 sentences of Godel's famous 1931 article (not famous > > enough, unfortunately.) > > YOU are TELLing US to READ something?? > I'm sorry, it doesn't work that way. > You ASK us a QUESTION about this, if you have read it. > We doubt this, frankly, since you obviously haven't understood it. How is that obvious? C-B |