How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)
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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
From: Transfer Principle on 2 Jul 2010 22:40 On Jun 30, 3:15 pm, "K_h" <KHol...(a)SX729.com> wrote: > "R. Srinivasan" <sradh...(a)in.ibm.com> wrote in message > > OK. Here I want ~Inf to be stated in the form that you > > mentioned, that is, every set is hereditarily finite. > Why do you think the axiom of infinity is false? What is > the basis for your belief in ~Inf? To me it is > self-evident that all the naturals exist. But there's a huge difference between "all the naturals" exist and "a _set_ of all naturals exists." To see the difference, try substituting "ordinals" for "naturals." I bet that K_h believes that all the ordinals exist, but not a _set_ of all ordinals (Burali-Forti). Notice that Srinivasan's base theory is NBG, which allows for proper classes. If all the naturals exist (as sets), then there is a class of all naturals. But it might be the case that this class is proper. Indeed, this appears to be _precisely_ the case in Srinivasan's own theory NBG-Infinity+D=0. In this theory, the class of all naturals is precisely the proper class of all ordinals. In short, omega=On.
From: Transfer Principle on 2 Jul 2010 23:00 On Jun 30, 11:34 pm, Ki Song <kiwisqu...(a)gmail.com> wrote: > Perhaps an analogy is in order! > I feel like what Charlie-Boo is asking people to do is analogous to > asking someone to perform the addition: > Sqrt{2}+Sqrt{3}, with a decimal precision of 10^(10^(10^100)) place. To me, there is one key difference. It appears that most posters, including Charlie-Boo (though not _all_ sci.math posters, but I digress), accept that in principle, given enough time and space, one can find sqrt(2)+sqrt(3) to that precision. After all, this is what is meant by computable -- there exists a Turing machine such that given any natural number, even 10^10^10^100 (often called a "googolplexplex," or a "googolduplex") as input, the machine prints that many digits of the computable number sqrt(2)+sqrt(3) and halts. But it's _not_ equally evident to Charlie-Boo that ZFC can prove the consistency of PA. I myself accept the proof of Con(PA) (though I still believe that the presence of epsilon_0 in any proof of Con(PA) is suspicious, I don't deny that the such a proof exists), but Charlie-Boo doesn't. What Charlie-Boo needs, therefore, is some evidence that convinces him that if one were to work out all the steps, one can eventually prove Con(PA) in ZFC, just as one can eventually find googolduplex digits of sqrt(2)+sqrt(3). So far, no poster or book has so convinced him. Therefore, there is no reason for him to believe that such a proof exists.
From: Transfer Principle on 2 Jul 2010 23:21
On Jul 1, 9:25 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jul 1, 9:01 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > or if he meant that was just a relative consistency > > proof he had referred to. > Anyway, Aatu is not saying just that there exists a relative > consistency proof nor just that, say, ZF or some other formal system > proves Con(PA), but rather he's saying that PA IS consistent. He's > saying that aside from whatever FORMAL proofs, PA is consistent - > PERIOD. His basis is for that is not a FORMAL proof, but rather his > conviction that the axioms of PA are true (and not even in confined to > a FORMAL model theoretic sense of truth, but rather that the axioms > are simply true about the natural numbers, as we (editorial 'we') > understand the natural numbers even aside from any formalization. But this raises an interesting point here. If Aatu can say that PA is consistent, _period_, without any formal proof whatsoever, then why can't Nguyen believe that PA is inconsistent, _period_, without formal proof? For that matter, why can't Herc believe that there exist only countably many reals, _period_, without formal proof, or Srinivasan believe that Infinity is false, _period_, without formal proof, or WM believe that certain large naturals don't exist, _period_, without formal proof? In reality, the only statements that posters are allowed to make, _period_, without formal proof, are those which ZFC proves, or at least are undecidable in ZFC. Any statement outright refuted by ZFC isn't allowed to be made, _period_, without formal proof. On the contrary, a formal proof will be demanded at best, with the poster who made the claim being asked to provide axioms, definitions, and so on. And at worst, the poster will be called a five-letter insult. So Nguyen is on the right track here. Aatu can state that PA is consistent because there's a theory T proves such -- even if Aatu states it without any such formal proof -- and that theory is ZFC. But no one is allowed to make a claim of anything refuted by ZFC without a formal proof in some (other) theory. This is a fact, no matter how much MoeBlee or anyone else may try to deny it. |