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 17:43 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. MoeBlee
From: K_h on 30 Jun 2010 18:15 "R. Srinivasan" <sradhakr(a)in.ibm.com> wrote in message news:46d58d89-34b1-40a9-a5a8-1ee250ba57e3(a)e5g2000yqn.googlegroups.com... On Jun 29, 8:33 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 29, 2:09 am, "R. Srinivasan" <sradh...(a)in.ibm.com> > wrote: > > > > ZF-"Inf'+"~Inf" > > > > That theory entails that every object is finite. And > > there is no > > definition of any infinite object possible in that > > theory. > > > OK. Here I want ~Inf to be stated in the form that you > mentioned, that is, every set is hereditarily finite. Why do you think the axiom of infinity is false? What is the basis for your belief in ~Inf? To me it is self-evident that all the naturals exist. _
From: Aatu Koskensilta on 30 Jun 2010 23:41 Chris Menzel <cmenzel(a)remove-this.tamu.edu> writes: > I believe the false claim above holds if we replace ZF with ZF-Inf+~Inf. Yes -- in fact PA and ZF-Inf+Inf (suitably formulated) are bi-interpretable. -- Aatu Koskensilta (aatu.koskensilta(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 00:14 On Jun 29, 5:23 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 29, 9:30 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > On Jun 29, 12:13 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > If you still can't see it, then, if I'm feeling generous, I'll outline it > > > for you. As to showing an exact sequence of primitive formulas of the > > > language of Z, no, that's just a chore. > > I don't know what you're referring to. I did ask for the statement of > > the theorem in ZFC, but nobody has come up with that either. > > So in summary, > > 1. ZFC can prove PA consistent - it's easy and lots of people have > > done it. > > 2. Nobody can give a reference to its being done. > > 3. Nobody can describe the proof that has been done in ZFC. > > 4. Nobody can give even the ZFC expression for the theorem itself. > > In other words, business as usual. > > 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, which the the smallest ordinal not reachable > from omega via finitely many additions, multiplications, and > exponentiations, but can be reached via finitely many _tetrations_ > since epsilon_0 = omega^^omega. This is why I so often mention > Ed Nelson and his proof attempt of ~Con(PA) involving tetration. Do you have a reference that shows a proof that PA is consistent, carried out entirely in ZFC? If so, what is the formal expression in ZFC that PA is consistent? How many lines are there in the proof? Which ZFC axioms are used that PA would need to carry out the proof entirely in PA? C-B
From: Charlie-Boo on 1 Jul 2010 00:17
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. Another question: How many ways can people propose something other than a reference to a proof in ZFC that PA is consistent? 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. (not mutually exclusive)etc. C-B > MoeBlee |