Models of set theory vs. models of first order theories
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From: Marc Alcobé García on 17 Feb 2010 04:13 On 17 feb, 09:55, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marc Alcobé García <malc...(a)gmail.com> writes: > > > In ZFC one can prove that given any finite list S of axioms of ZFC > > there is a transitive SET where that S is true. > > > But we also have a corollary of the compactness theorem saying that a > > theory T has a model iff every finitely axiomatized part of T has a > > model. If this corollary was provable within ZFC then ZFC would prove > > its own consistency via the completeness theorem... > > The corollary is provable in ZFC. We have that ZFC proves, for each of > its finite subtheory T, "T is consistent". But ZFC does not prove "every > finite subtheory of ZFC is consistent". > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus That's why Kunen says that platonistically we have a "proof" of the consistency of ZFC, since although we cannot prove "every finite subtheory of ZFC is consistent" we "know" that this is already the case?
From: Aatu Koskensilta on 17 Feb 2010 04:49 Marc Alcob� Garc�a <malcobe(a)gmail.com> writes: > That's why Kunen says that platonistically we have a "proof" of the > consistency of ZFC, since although we cannot prove "every finite > subtheory of ZFC is consistent" we "know" that this is already the > case? No. Platonism doesn't really enter into it, but if we regard the mathematical picture of the world of sets as envisaged in the cumulative hierarchy story as clear and compelling, and the axioms of ZFC as justified on that picture, the consistency of ZFC trivially follows -- the axioms are all true, the rules of inference preserve truth, hence no falsity (such as a contradiction) is provable. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: LauLuna on 18 Feb 2010 19:18 On Feb 3, 10:18 pm, Marc Alcobé García <malc...(a)gmail.com> wrote: > Shoenfield and Levy agree in saying that set theory is but an > investigation of what is meant by the "and so on" of this particular > collection of individuals: > {}, {{}}, {{},{{}}}, and so on. This is exactly what I have in mind > when I think of V. However it is clear that it does not seem to > satisfy the needs of first order semantics to be a model of anything. That's a very interesting way of putting it. Since I am no Platonist, I'd say the answer is to be expected from a theory of our capability of constructing or defining mathematical objects; perhaps phenomenology, perhaps linguistics, perhaps either. In my opinion, the inexhaustibility of the mathematical universe expressed by the 'and so on' is just the capability of our reason to go beyond whatever has been objectified: the function 'set of' is essentially the function 'thought of'. I know I'm sounding a bit enigmatic. Sorry. It's late.
From: Marc Alcobé García on 19 Feb 2010 03:32 > > Shoenfield and Levy agree in saying that set theory is but an > > investigation of what is meant by the "and so on" of this particular > > collection of individuals: > > {}, {{}}, {{},{{}}}, and so on. This is exactly what I have in mind > > when I think of V. However it is clear that it does not seem to > > satisfy the needs of first order semantics to be a model of anything. > > Since I am no Platonist, I'd say the answer is to be expected from a > theory of our capability of constructing or defining mathematical > objects; perhaps phenomenology, perhaps linguistics, perhaps either. Maybe you would enjoy reading Kai Hauser's "Gödel's program revisited. Part I The turn to phenomenology" where he "explores ideas from phenomenology to specify a meaning for his [Gödel's] mathematical realism that allows for a defensible epistemology". This point of view reminds me of that of Yvorra, expressed in his introduction to his lecture notes "Lógica y Teoría de conjuntos", although I think he talked about theory of knowledge rather than phenomenology.
From: Marc Alcobé García on 19 Feb 2010 08:33
On 17 feb, 09:55, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marc Alcobé García <malc...(a)gmail.com> writes: > > > In ZFC one can prove that given any finite list S of axioms of ZFC > > there is a transitive SET where that S is true. > > > But we also have a corollary of the compactness theorem saying that a > > theory T has a model iff every finitely axiomatized part of T has a > > model. If this corollary was provable within ZFC then ZFC would prove > > its own consistency via the completeness theorem... > > The corollary is provable in ZFC. We have that ZFC proves, for each of > its finite subtheory T, "T is consistent". But ZFC does not prove "every > finite subtheory of ZFC is consistent". > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus I still have some trouble here. The corollary to the compactness theorem says in the metatheory that: T' has a model iff every finitely axiomatized part T of T' has a model. We have, for each finitely axiomatized part T of ZFC: |-_ZFC Con(T). This is a statement in the metatheory about a formal theory [ZFC] and a statement in that formal theory [Con(T)]. If ZFC is now regarded as a formalization of the metatheory, then Con(T) is a theorem in the metatheory for each T. So, for each T we have a model. I don't see why from this we cannot conclude in the metatheory that T' (i. e. ZFC) has a model... |