Godels theorem ends in paradox-simply proof
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From: George Greene on 5 Jun 2010 00:02 On Jun 4, 12:04 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Gödel's proof establishes that for a consistent formal theory T Well, yeah, but since nobody is going to prove that T is consistent, do we care?? > there's a Pi-1 sentence P that's true but unprovable in T. This is a misuse of "true". That sentence is true IN THE MODEL OF T THAT WAS PRESUMED TO EXIST WHEN T WAS PRESUMED CONSISTENT. The fact that that sentence is not going to be provable in T means that THERE MUST ALSO EXIST MODELS OF T IN WHICH THAT SENTENCE IS *FALSE*. Therefore, THAT SENTENCE IS NOT "true". > That is, P has the > form "for all naturals n, Q(n)" No, it doesn't. PA can't tell what's natural and what isn't.
From: Daryl McCullough on 5 Jun 2010 09:39 George Greene says... > >On Jun 4, 12:04=A0pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> G=F6del's proof establishes that for a consistent formal theory T > >Well, yeah, but since nobody is going to prove that T is consistent, >do we care?? > >> there's a Pi-1 sentence P that's true but unprovable in T. > >This is a misuse of "true". That sentence is true IN THE MODEL >OF T THAT WAS PRESUMED TO EXIST WHEN T WAS PRESUMED CONSISTENT. >The fact that that sentence is not going to be provable in T means >that THERE MUST ALSO EXIST MODELS OF T IN WHICH THAT SENTENCE IS >*FALSE*. If the T in question is PA, then there is an *intended* model, which is the naturals. The Godel sentence is true in *that* model. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 5 Jun 2010 11:16 Daryl McCullough wrote: > George Greene says... >> On Jun 4, 12:04=A0pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>> G=F6del's proof establishes that for a consistent formal theory T >> Well, yeah, but since nobody is going to prove that T is consistent, >> do we care?? >> >>> there's a Pi-1 sentence P that's true but unprovable in T. >> This is a misuse of "true". That sentence is true IN THE MODEL >> OF T THAT WAS PRESUMED TO EXIST WHEN T WAS PRESUMED CONSISTENT. >> The fact that that sentence is not going to be provable in T means >> that THERE MUST ALSO EXIST MODELS OF T IN WHICH THAT SENTENCE IS >> *FALSE*. > > If the T in question is PA, then there is an *intended* model, > which is the naturals. The Godel sentence is true in *that* > model. Iirc, somewhere in the forum, AK said something to the effect that the naturals collectively isn't a model (something like the truths about the naturals aren't model theoretically truths). Would you be able to spell out "that" model? And to the extend a language model must decide the truth value of a formula, would (1) be true in "that" model? [(1) was defined in the other thread you were also in. If requested I'll repost the formula here.]
From: Daryl McCullough on 5 Jun 2010 18:09 Nam Nguyen says... > >Daryl McCullough wrote: >> If the T in question is PA, then there is an *intended* model, >> which is the naturals. The Godel sentence is true in *that* >> model. > >Iirc, somewhere in the forum, AK said something to the effect that >the naturals collectively isn't a model (something like the truths >about the naturals aren't model theoretically truths). I'm not sure what he meant by that. >Would you be able to spell out "that" model? There's not much to spell out. The naturals consists of a starting element, zero, and a one-to-one function successor such that zero is not the successor of anything. That's really all there is to know about the naturals: it's the smallest set containing zero and closed under the successor function. That's the U for the model. For the interpretation of plus, let P be the smallest set of triples of naturals such that 1. for each natural number x, <0,x,x> is in P 2. for each triple of natural numbers x, y and z, if <x,y,z> is in P, then <S(x),y,S(z)> is in P. For the interpretation of times, let T be the smallest set of triples such that 1. for each natural number x, <0,x,0> is in T. 2. for each quadruple of natural numbers x,y, w, and z, if <x,y,z> is in T, and <z,y,w> is in P, then <S(x),y,w> is in T. That's a model for the language of arithmetic. >And to the extend a language model >must decide the truth value of a formula, would (1) be true in "that" >model? [(1) was defined in the other thread you were also in. If >requested I'll repost the formula here.] I don't remember what statement you're talking about, but a model decides every closed formula in its language. That doesn't mean that *I* know the truth of every formula. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 7 Jun 2010 14:28
Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes: > herbzet <herbzet(a)gmail.com> writes: > >> Of course, it is a short inference from "neither S nor ~S is provable" >> to "There is a true statement that is not provable" if one assumes, as >> classical logic does (but intuitionist logic does not) that one of S >> or ~S is true. > > Gödel's proof establishes that for a consistent formal theory T there's > a Pi-1 sentence P that's true but unprovable in T. That is, P has the > form "for all naturals n, Q(n)" with Q a decidable predicate, and if T > is consistent, Q in fact holds of all naturals and P is unprovable in > T. The proof is perfectly constructive -- "logic-free" even, in > technical jargon -- so there's no need to invoke classical logic. But it ends in meaninglessness, right? -- Jesse F. Hughes "Knowing about logic is not the same as being in touch with reality." -- David Kastrup |