Torkel Franzen on truth
in [Logic]
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From: george on 7 Dec 2007 17:26 On Dec 6, 5:54 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > (It is one anyone > teaching this stuff stresses: sure, we need to assume that the system > we are dealing with is indeed consistent if we are get to see that the > canonical Gödel sentence is true on the standard interpretation.) No, you don't. You DON'T see that the system is consistent because you can't prove it, period. You don't know that the standard interpretation or any other interpretation EVEN EXISTS, because if one did, it would, by existing, prove consistency. Everything you think you want to say about "the standard interpretation" is something you actually wind up proving in ZFC after invoking the (minimal) set satisfying the axiom of infinity.
From: Alan Smaill on 7 Dec 2007 17:45 george <greeneg(a)cs.unc.edu> writes: > What DOES matter and what IS the point is that a "cuonterexample > to FLT" is going to be THE *n* and NOT the "z", as klept0 was mis- > alleging. > > And it also matters that you have to have some sort of Pi-1 definition > of > exponentiation to get this into the language of PA. I presume the > original > Godel proof uses the chinese remainder theorem among other things to > manage that. Yes it does. -- Alan Smaill
From: Peter_Smith on 7 Dec 2007 17:59 On 7 Dec, 22:06, george <gree...(a)cs.unc.edu> wrote: > On Dec 6, 5:54 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > Hold on, hold on. You've rather ripped this out of context from Dan's > > lecture notes. He two paragraphs earlier introduces the assumption > > that we are dealing with systems of arithmetic S which are *sound* > > (and hence consistent). > > Come ON. That is a lame dodge on YOUR part. Well, I think not. But it isn't my job to defend someone else's informal lecture notes: so while I still think you are misreading Dan, let's agree to differ about the correct reading of his notes. > > the > > canonical Gödel sentence will indeed be true on the standard > > interpretation. > > But proving that is quite beyond the scope of the material. > You have to go somewhere like ZF and epsilon_0-induction to prove > THAT. But again, what *I* said was that the canonical Gödel sentence *of a sound theory* will be true on the standard interpretation. That follows immediately, without going somewhere like ZF, from 1) For any theory S, the canonical Gödel sentence G_S is true on the standard interpretation if and only if S doesn't prove G_S. [By construction] and 2) For any sound theory S, S doesn't prove G_S [By Gödel's semantic argument for incompleteness]
From: Nam D. Nguyen on 7 Dec 2007 23:27 george wrote: > On Dec 7, 4:44 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> Well, I'm mainly taking issue with the claim that "we do not >> have the foggiest idea if PA is consistent". That's not true. > > Who is claiming that? TF? I never saw him entertain that seriously > in life. > The book just says that "in general", right? > Not for PA specifically, right? > > In any case there do exist consistency proofs for PA in systems of set > theory > that do seem more than just "foggily" sound. So we do have an idea. > It's just not the kind of idea that deserves any sort of mathematical > respect. > > My personal opinion is that mathematical respect flows out of the > completeness theorem: IF it is inconsistent, THEN THAT MUST be > provable. Why "must", a _subjective_ verb? What happens if such inconsistency proof is beyond human reach? > Therefore, the BURDEN of proof rests ALWAYS UPON people expressing > doubts about consistency. But what happens if the theory is genuinely consistent but it's *impossible* to know that? And if it's impossible to know the consistency then isn't it true we'd not know the state of inconsistency, and therefore might harbor the doubt that it might be inconsistent, logically speaking? > If their doubts are at all justified then a proof must in fact exist, > so they are obligated to bring it back alive -- that demand is > reasonable if their position is justifiable. "Doubt" doesn't have to follow dualism-rule: I could doubt *both* opposite claims simply because I might not know enough about any of them. For instance, I doubt if ~GC is provable in Q and I also doubt GC is provable in Q! (Odd huh? But could one do better than I've done?)
From: abo on 8 Dec 2007 00:53
On Dec 7, 10:44 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > pbopa...(a)gmail.com says... Who was me - one of my sons had changed my login on my computer. Grrr... > > > > >On Dec 7, 5:17 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > >> In any case, whether or not we can be said to *know* with > >> certainty that PA is consistent, it is certainly false to > >> say that "we do not have the foggiest idea if PA is consistent". > >> That's completely wrong. The consistency of PA is as certain > >> as *anything*. > > >Well, it seems to me that this last assertion is completely wrong. "2 > >+ 2 = 4" seems more certain to me than the consistency of PA, and I > >imagine the same is true for you, as well. > > Well, I should probably say as certain as any nontrivial > mathematics. That's better. On the other hand, there's still a difference between, "There's a prime number between 10 and 20" and "For every natural number n there's a prime number greater than n." > I don't think it is possible to do anything > nontrivial without PA (or something equivalent). Do you consider Quadratic Reciprocity or Bertrand's Postulate non- trivial? Both can be proved without using PA, that is in second-order PA \ {successor axiom}, which has as a model all initial segments as well as the standard model. I suspect Fermat's Last Theorem (which is surely non-trivial?) can be proven in this reduced theory as well. > > >After all, the consistency of PA - that a particular > >logico-mathematical system does not ever produce among > >its deductions "not 0 = 0" - presumably depends on an > >argument for you to believe it. "2 + 2 = 4" does not. > > Sure it does. Really? You only believe "2 + 2 = 4" because of an argument? That is, I'm not denying that you *can* use an argument (a Principia Mathematica-style proof) to arrive at "2 + 2 = 4". I'm saying that's not the actual, causal reason why you believe it. Of course, you might be different from me in this matter, but I would be surprised if you are (and quite interested if you are!). > |