How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Charlie-Boo on 29 Jun 2010 10:16 On Jun 29, 9:16 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > billh04 <h...(a)tulane.edu> writes: > > Are you saying that it is a theorem of ZFC that PA is consistent? > > Sure. That is, the statement "PA is consistent" formalized in the > language of set theory as usual What is that formal expression? C-B > is formally derivable in ZFC (and > already in much weaker theories). > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Charlie-Boo on 29 Jun 2010 10:33 On Jun 29, 10:22 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > What is that formal expression? > > To find out you need to read a logic book. It appears the generous > explanations various people have provided for your benefit in news are > not sufficient. Sorry, but I'm not asking for an explanation of anything. I am asking for the formal expression - a string of characters - that you referred to. Are you able to quote it (or write it yourself)? Are you saying that you can't even express the theorem in the first place? C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Charlie-Boo on 29 Jun 2010 10:35 On Jun 29, 10:22 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > What is that formal expression? > > To find out you need to read a logic book. It appears the generous > explanations various people have provided for your benefit in news are > not sufficient. Explanations of what? I just asked for the name of the book or article that you and they are referring to as having a proof of PA consistency carried out in PA. What is the book or article title? C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Chris Menzel on 29 Jun 2010 10:34 On Tue, 29 Jun 2010 03:34:12 -0700 (PDT), Charlie-Boo <shymathguy(a)gmail.com> said: > On Jun 29, 12:18 am, 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 > > There are too many sets to map them 1-to-1 with the natural numbers. Apparently you have yet to master the semantic role of adjectives. To say nothing of basic set theory.
From: Charlie-Boo on 29 Jun 2010 10:47
On Jun 29, 10:34 am, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Tue, 29 Jun 2010 03:34:12 -0700 (PDT), Charlie-Boo > <shymath...(a)gmail.com> said: > > > > > > > On Jun 29, 12:18 am, 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 > > > There are too many sets to map them 1-to-1 with the natural numbers. > > Apparently you have yet to master the semantic role of adjectives. Ok, then tell me. What is the semantic role of adjectives? C-B http://blog.mrm.org/wp-content/uploads/2007/09/wizardofoz.jpg > To > say nothing of basic set theory.- Hide quoted text - > > - Show quoted text - |