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: Charlie-Boo on 29 Jun 2010 10:06 On Jun 29, 8:58 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > How about when you said that Gentzen proved PA consistent using ZFC? > > No one ever proves anything using ZFC in the sense of producing formal > derivations. What about the MetaMath web site - or do you think it's BS as some do? > Rather, they prove mathematical theorems using mathematical > principles formalized in ZFC. By certain elementary considerations we > know the formalizations of such theorems are formally derivable in ZFC. Ok. What is the reference to the proof in ZFC of PA consistency doing it that way? 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 10:15 On Jun 29, 9:14 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > On Jun 28, 8:58 am, Frederick Williams <frederick.willia...(a)tesco.net> > > wrote: > > >> Yes, you can: take Gentzen's proof (or Ackermann's etc) and formalize > >> it in ZFC. > > > Give the slightest bit of details. > > Read a logic book. What is the title of a book that includes it? > >> It has everything to do with V_omega. > > > That's not a ZFC axiom. > > Indeed. It is a set. What are the essential ZFC axioms that are used that PA would need to carry out the proof? C-B > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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 |