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 14 Jun 2010 11:42 ZFC/PA is supposed to do all ordinary mathematics*. But it is easy to prove that PA is consistent (its axioms and rules preserve truth) yet (by Godel-2) PA can't do such a simple proof as that. Since PA can't prove something as simple as that, how could anyone be so stupid as to claim ZFC/PA is a good basis for all of our ordinary math? C-B Answer: I failed to "Never underestimate the stupidity of academia." * Putting aside the fact that this ill-defined, ridiculous association of the temporal with the timeless is a pitiful attempt by the Prof.s of the world to yet again say they have captured all of Math despite Godel. (Hilbert Lives!)
From: MoeBlee on 14 Jun 2010 11:48 On Jun 14, 10:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > ZFC/PA is supposed to do all ordinary mathematics*. The common claim is that ZFC axiomatizes all (or virtually all) ordinary mathematics. It is not claimed that PA axiomatizes all (hor even virtually all) ordinary mathematics. MoeBlee
From: George Greene on 14 Jun 2010 23:04 On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > ZFC/PA There is no such hybrid beast. PA is one thing. ZFC is another. Theories "of this type" cannot prove THEIR OWN consistency (so PA cannot prove that PA is consistent, and ZFC cannot prove that ZFC is consistent). A stronger theory can often prove the consistency of a weaker one. ZFC *can* (and does) prove that PA is consistent: the set required to exist by ZFC's Axiom of Infinity is (the domain of) a model for PA.
From: Charlie-Boo on 24 Jun 2010 13:45 On Jun 14, 11:04 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > > > ZFC/PA > > There is no such hybrid beast. ZFC includes PA. C-B > PA is one thing. > ZFC is another. > Theories "of this type" cannot prove THEIR OWN consistency > (so PA cannot prove that PA is consistent, and ZFC cannot prove that > ZFC is consistent). > A stronger theory can often prove the consistency of a weaker one. > ZFC *can* (and does) prove that PA is consistent: the set required to > exist by ZFC's Axiom of Infinity is (the domain of) a model for PA.
From: Charlie-Boo on 24 Jun 2010 13:47
On Jun 14, 11:45 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Charlie-Boo <shymath...(a)gmail.com> writes: > > Since PA can't prove something as simple as that, how could anyone be > > so stupid as to claim ZFC/PA is a good basis for all of our ordinary > > math? > > Who makes this claim? MoeBlee > You're hallucinating. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |