Godels theorem ends in paradox-simply proof
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From: Nam Nguyen on 7 Jun 2010 22:55 Daryl McCullough wrote: > 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. Did you mean something like the predicates for (S,+,*,<) are all finite sets therefore aren't worth our effort to spell them 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 doesn't sound like spelling imho. It's like I requested you to spell out "Alphabet", and you said it starts with "A" and there are some number of alphabets after that! > 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. I guess one should be precise in these matters. If you still had to explain anything about '+', "*' below then perhaps you should have been a bit more careful in saying "That's really all there is to know about the naturals"! > > 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. Aren't these a bit circular: you're explaining the naturals using the naturals? > >> 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, it is: > pGC <-> "There are infinitely many examples of GC" > cGC <-> "There are infinitely many counter examples of GC" > (1) pGC xor cGC > but a model > decides every closed formula in its language. By definition of a model yes. > That doesn't mean > that *I* know the truth of every formula. But no one has asked for the the truth of every formula. I just requested for only _1_ formula: (1)!
From: Marshall on 7 Jun 2010 23:02 On Jun 7, 7:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Daryl McCullough wrote: > > > 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. > > Aren't these a bit circular: you're explaining the naturals using > the naturals? I guess we can add "circularity" to the list of things that you don't understand about definitions. Marshall
From: Nam Nguyen on 7 Jun 2010 23:10 Marshall wrote: > On Jun 7, 7:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Daryl McCullough wrote: >> >>> 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. >> Aren't these a bit circular: you're explaining the naturals using >> the naturals? > > I guess we can add "circularity" to the list of things that you > don't understand about definitions. 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. > 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. Apparently Marshall couldn't recognize circularity when he actually saw it!
From: Marshall on 8 Jun 2010 09:52 On Jun 7, 8:10 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Jun 7, 7:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Daryl McCullough wrote: > > >>> 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. > >> Aren't these a bit circular: you're explaining the naturals using > >> the naturals? > > > I guess we can add "circularity" to the list of things that you > > don't understand about definitions. > 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. > > > 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. > > Apparently Marshall couldn't recognize circularity when he actually saw it! Apparently Nam hallucinates circularity when it's not there! Marshall
From: Aatu Koskensilta on 8 Jun 2010 21:19
herbzet <herbzet(a)gmail.com> writes: > If you're not disagreeing with anything, I'm not sure what you're > driving at. Could you elaborate? I was just pointing out that while your reasoning was perfectly fine there's no need to invoke classical logic to conclude from G�del's proof that for every sufficiently expressive consistent formal theory there's an arithmetical truth it doesn't prove. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |