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 14:25 On Jun 30, 12:36 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> wrote: > On Jun 30, 7:42 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 29, 10:45 pm, "R. Srinivasan" <sradh...(a)in.ibm.com> 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 > > > I think it is not known whether this proposition (That there are sets > > > other than the hereditarily finite sets) is undecidable, refutable, or > > > provable in ZF-I. Undecidability of this propostion is just an > > > assumption as far as I know. > > > As far as you know. > > Yes. Undecidability of that proposition is a mere assumption. You may say that the consistency of ZF is a mere assumption. But if we I assume that ZF is consistent (I may leave out mentioning that background assumption elsewhere in this context): I have a hunch it's provable in Z that ZF-I neither proves nor refutes "there are sets that are not hereditarily finite". ZF-I does not refute "there are sets that are not hereditarily finite". And I have a hunch that in Z we can prove that ZF-I does not prove "there are sets that are not hereditarily finite". > At my age (53) and given my day job, I have to be conservative about > what I read. I do not want to fill my head with stuff that I do not > believe in. So you've NEVER read a book on set theory and/or mathematical logic? > For what it is worth: Your paradigm WHAT paradigm? Where did I announce I have a certain paradigm? > is based on the philosophy called > Platonism, Whatever my philosophy, it's not platonism. It's obvious you've not read the posts I've made about the subject. > which is the only way out for you to deny circularity. WHAT circularity? MoeBlee
From: MoeBlee on 30 Jun 2010 14:35 On Jun 30, 1:27 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > "R. Srinivasan" <sradh...(a)in.ibm.com> writes: > > If I had wasted my time trying to dig into the rubbish that you have > > laid out above, I would not have had much time or energy left to deal > > with the kind of stuff that *i* consider worth doing. > > You're of course free to spend your time and energy however you > choose. But why do you think others should take any notice of your > interests and inclinations? NAFL revolutionizes logic! NAFL finally releases mankind from the chokehold of Platonist orthodoxy! Sheesh, I thought you were keeping up with this. MoeBlee
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 |