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 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 22:41 On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > ZFC/PA You're AN IDIOT. ZFC IS ONE thing. PA IS ANOTHER, SIMPLER thing. ZFC is A SET theory. PA is a theory OF ARITHMETIC. First-order theories in general cannot prove THEMSELVES consistent, so first-order PA cannot prove that first-order PA is consistent. First-order ZFC, however, IS A STRONGER THEORY (it has an axiom of infinity), SO IT CAN AND DOES prove that PA is consistent. Your problem is that you presumed to talk about ZFC/PA like it was one thing. Your problem is that you flaunted the fact that YOU ARE IGNORANT OF THE RELEVANT DIFFERENCES BETWEEN THE TWO. ZFC is a set theory. PA is a number theory. PA doesn't know what an infinite set is. ZFC does. That is the main reason why ZFC can prove that PA is consistent (a model of PA *has* to be infinite, and PA can't prove that anything is infinite, since in its standard model, NOTHING IS).
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
From: Charlie-Boo on 24 Jun 2010 14:01
What does ZFC/PA really provide (mathematically and psychologically)? PA is a programming language - an enumeration of the r.e. sets (recursive objects.) And the ZFC part is the DATA STRUCTURES of this "programming language". When programmers need to go beyond aleph-1 integers they use local arrays or built-in structures like representations of a subset of the real numbers. We need an infinite number of variable names. So we call it an array using one name and an infinite number of subscripts. (It's really unbounded as far as we know but finite.) So what does a programming language with data structures provide you? You can run a program and maybe see that it halts. And if it halts, we have a proof that it halts. And when a program halts, what does that prove? There are a lot of things it could prove (e.g. Fermat and Goldbach) so some people say it can do most anything we can do. But the Theory of Computation tells you there are some questions that even a solution (oracle) to the Halting Problem can't provide. But I guess they would just claim that this isn't "ordinary mathematics". *sigh* C-B On Jun 14, 11:42 am, Charlie-Boo <shymath...(a)gmail.com> wrote: > 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!) |