How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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From: Chris Menzel on 2 Jul 2010 05:34 On Thu, 1 Jul 2010 08:41:44 -0700 (PDT), Charlie-Boo <shymathguy(a)gmail.com> said: > On Jul 1, 2:04 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> Charlie-Boo <shymath...(a)gmail.com> writes: >> > To see it means to have actually created it, and its actual creation >> > would answer the very interesting question of whether ZFC can prove >> > that PA is consistent even though PA can't. >> >> We already know the answer. The axioms used in the proof are a >> restricted form of comprehension, infinity, union and pairing. > > Why do we need each of these? Of course infinity = PA = arithmetic. > But why are the others needed? These are the questions you could answer for yourself with just a bit of study. The tragic thing is, these topics are so marvelously interesting and yet you insist on dwelling in ignorance. (Infinity = PA? So sad...)
From: Chris Menzel on 2 Jul 2010 05:40 On Wed, 30 Jun 2010 21:23:23 -0700 (PDT), Charlie-Boo <shymathguy(a)gmail.com> said: > On Jun 29, 9:51 pm, herbzet <herb...(a)gmail.com> wrote: >> "Jesse F. Hughes" wrote: >> > Of course, I'm really here for lower entertainment. I want posts about >> > the Hammer, about how surrogate factoring moves the stock market, about >> > the most influential mathematicians on the planet. But still I pretend >> > to care about arguments, if only for appearance's sake. >> >> Well, I didn't come here for the low comedy -- that's simply not an idea >> that had occurred to me. But, it being made explicitly an option -- I >> suppose it's a valid choice, even one of some value. >> >> If I have any reservation, it would just be that I don't think that >> encouraging crankery for its entertainment value > > crank: "an unbalanced person who is overzealous in the advocacy of a > private cause" You're talking about you and CBL, right?
From: Frederick Williams on 2 Jul 2010 10:39 Charlie-Boo wrote: > > On Jul 1, 3:50 pm, Frederick Williams <frederick.willia...(a)tesco.net> > wrote: > > Charlie-Boo wrote: > > > No, PA and ZFC both have Peano's Axioms. > > > > What formulations of first order PA and ZFC have _any_ of their > > non-logical axioms in common? None. Why? Well, for starters, all of > > the proper axioms of ZFC have the predicate symbol $\in$ in them, none > > of PA's axioms do because $\in$ is not even in the language of PA > > > > > The only difference is the > > > universal set, which is N in PA > > > > There are no sets, universal or otherwise, in first order PA. > > What does (existsX)P(X) mean? It means that there's a natural number which has property P. -- I can't go on, I'll go on.
From: MoeBlee on 2 Jul 2010 17:21 On Jul 1, 5:52 pm, herbzet <herb...(a)gmail.com> wrote: > MoeBlee wrote: > > > CORRECTION below: > > > On Jul 1, 5:10 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > However, even though the definition I gave is okay, and provides these > > > theorems, > > > > a theory T is inconsistent <-> T proves a contradiction > > > > a set of axioms X for a theory T is inconsistent <-> X proves a > > > contradiction > > > Oops, delete that. > > Not sure if "that" is the previous sentence, or the previous 2 sentences. Good point. Just the previous sentence. > > > a theory T that is axiomatized by X is inconsistent <-> X proves a > > > contradiction, > > > > I realize that in my own notes I actually do use a different > > > definition, which, again, is equivalent for THEORIES but different for > > > arbitrary sets of formulas: > > Missing the nuance here, unless you are distinguishing between > a set Gamma of formulas *implying* both P and ~P, and a set Gamma > of formulas *containing* both P and ~P. Lets use 'derives' instead of 'implies' ('entails') so that we're clearly purely syntactical. But, yes, my first definition was not as good. Better to say that a set of formulas G is inconsistent iff we can derive a contradiction from G. > > > a set of formulas G is consistent <-> there is no formula P such that > > > P and ~P are provable from G > > OK, so far so good. > > > > a set of formulas G is inconsistent <-> G is not consistent > > Can't object to that. > > > and I think that is an easier and nicer definition to work with as it > > gives all at once: > > > a set of formulas G is inconsistent <-> G proves a contradiction > > > a set of formulas G is inconsistent <-> G proves every formula (in the > > language) > > No doubt missing some of the issues here, but: > > For classical logic the following sets are extensionally equivalent: > > 1) a set of formulas G that proves both P and ~P for some formula P. > 2) a set of formulas G that proves a contradiction, i.e., some formula (P & ~P). > 3) a set of formulas G that proves every formula (in the language). Exactly. You're agreeing with me. > For non-classical logics, these may diverge. Of course. > > What (3) describes is called a trivial theory. We don't necessarily > care so much about inconsistency -- a paraconsistent logic will tolerate > some inconsistency of type (1) or (2) in a theory T without T's necessarily > becoming trivial. > > We don't worry so much about inconsistency per se -- we worry about > triviality instead. From a paraconsistent viewpoint that's true. So whether we adopt: consistent <-> derives a contradiction or consistent <-> there's a formula not derivable would determine how the paraconsistent advocate would couch his views as to consistency, as he would not mind a theory with contradictions as long as the "explosive property" were absent so that contradictions don't derive everything. I think we are in accord, as far as I can tell. MoeBlee
From: MoeBlee on 2 Jul 2010 17:24
On Jul 2, 2:31 am, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > Peter Smith's wonderfully lucid book _An Introduction to Gödel's > Theorems_î highly recommended for this. But make sure to get a newer printing, since he made a number of important changes. MoeBlee |