Torkel Franzen on truth
in [Logic]
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From: Daryl McCullough on 19 Dec 2007 09:27 In article <c252f3f9-e5b9-42fd-a79e-bc25f4f5cffd(a)d4g2000prg.googlegroups.com>, Newberry says... > >On Dec 17, 6:37 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >So noted. I also note that most people do not find Gentzen's proof >> >very convincing. And I still do not see how a proof in a stronger >> >system could be more convincing than a proof in the system itself. >> >> The *strength* of the system is not relevant so much as whether the >> axioms are themselves intuitively true. A proof in a theory whose >> axioms are intuitively true is more useful and interesting than >> a proof in a theory whose axioms are not intuitively true. >> >> So, for example, a proof in PA + the negation of Goldbach's >> conjecture would not be very convincing, because we have no >> reason to believe that the negation of Goldbach's conjecture >> is true. > >Why is a consistency proof of a theory in a stronger theory >interesting? Didn't I just answer that? >> The *strength* of the system is not relevant so much as whether the >> axioms are themselves intuitively true. A proof in a theory whose >> axioms are intuitively true is more useful and interesting than >> a proof in a theory whose axioms are not intuitively true. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 19 Dec 2007 09:32 Newberry says... > >On Dec 17, 6:54 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >Let me see if I can summarize your position. >> >a) The human neural system does not surpass the Turing machine. >> >> Yes, I believe that. > >The same way you could prove that there is no free will. I never said I could do that, did I? I'm not sure what free will really means, so I won't venture to say that there is or is not such a thing. >Let me make sure that I understand the whole thing. By the heuristic >learning algorithms we have arrived at the conclusion that the axioms >of ZFC are true and also that ZFC is consistent. (What is the degree >of certainty, 100%?) In additions there is a also a consistency proof >of ZFC in a stronger system (ZFC + an axiom of infinity. (What is the >cogency of this proof?) I don't know what you mean by "cogency". As I have said, a proof is only interesting if you learn something from it. If the conclusion is a nonobvious consequence of the axioms. It's a little bit interesting that the existence of large cardinals implies the consistency of ZFC, but it's not difficult to see why. The reason I believe that ZFC is consistent is certainly not because of large cardinal axioms, it's because I find the cumulative hierarchy picture of the construction of the set-theoretic universe to be compelling. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 19 Dec 2007 09:35 Aatu Koskensilta says... > >On 2007-12-15, in sci.logic, Daryl McCullough wrote: >> Whether I "know" that or not depends on what you mean by "know". >> For practical purposes, you know something if you believe it and >> there are goods reasons for believing it and believing otherwise >> seems difficult. That's the case with ZFC. I don't have any absolute >> argument for why it is consistent, it's just that it seems very unlikely >> to me that it could be inconsistent and to have that inconsistency >> undiscovered before now. > >Why? We don't have any good data on which to judge the likelihood of that. >Happily, as you later note, we have much better grounds to think ZFC is >consistent, sound for arithmetical statements, and so on. The reasons I personally believe that ZFC is consistent is because of the intuitively clear picture of the set-theoretic universe given by the cumulative hierarchy. It's very hard to imagine what could go *wrong*. -- Daryl McCullough Ithaca, NY
From: Aatu Koskensilta on 19 Dec 2007 09:40 On 2007-12-19, in sci.logic, Daryl McCullough wrote: > The reasons I personally believe that ZFC is consistent is because > of the intuitively clear picture of the set-theoretic universe given > by the cumulative hierarchy. It's very hard to imagine what could > go *wrong*. Right. I was just noting that the "we haven't yet run into an inconsistency" argument is a very weak one, and that, in fact, as you did note, we do have a much better reason for believing ZFC consistent, sound for arithmetical statements, and so on. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: abo on 19 Dec 2007 10:06
On Dec 19, 2:40 pm, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > On 2007-12-15, in sci.logic, abo wrote: > > > Secondly, are any proofs of PA's consistency "interesting or useful"? > > Sure. From Gentzen's consistency proof we learn, for example, that if PA > proves a statement of the form "AxEyP(x,y)" with P decidable, there is a > function F, below the epsilon-0'th level of a hierarchy of fast growing > recursive functions, such that AxP(x,F(x)). > > > Or is it only when a theory proves its own consistency that it is not > > interesting or useful? > > Given that inconsistent theories prove their own consistency, if we merely > know of some theory T that it proves its own consistency nothing interesting > at all can be concluded. That's pretty vacuous. Given that inconsistent theories prove "PA is consistent", if we merely know of some theory T that it proves "PA is consistent" nothing interesting at all can be concluded. > If, inspecting the proof, we find that only > principles we accept as correct are in fact used in the proof, its then of > course useful in convincing us of the consistency of T. For most theories > we're interested we know this can't happen, since the conditions of the > second incompleteness theorems apply. > Yes, well I was interested in Daryl's definitive statement that no such proof could be interesting or useful. Apparently he meant it only to apply to theories where the conditions of the incompleteness theorem 2 apply, in which case I can understand why he thinks that there are proofs of "PA is consistent" which are interesting and useful, but not proofs (in T) of "T is consistent." |