Models of set theory vs. models of first order theories
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From: Aatu Koskensilta on 15 Feb 2010 18:51 Marc Alcob� Garc�a <malcobe(a)gmail.com> writes: > This suggests a question: what necessary and sufficient condition must > a theory meet to be able to formalize consistency statements in it? We need just a bit of arithmetic. > I think that to answer this question one must show that there is a > formal theory T where one is able to define the notions of recursive > or definable sets, of consistent sets of axioms, etc... Let me call > that theory T a metatheory. I would expect a metatheory to be a theory > where any other theory can be interpreted. There is no "metatheory" in this sense, if we require a metatheory to be consistent. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Marc Alcobé García on 16 Feb 2010 03:55 Returning to my original concerns. Kunen (Ch. IV §2 Relativization) defines (2.2): (a) "phi is true in M" as an abbreviation for the sentence phi relativized to M, where M is any class (a formula M(x,v) where x is a free variable and v a parameter). (b) "S is true in M" or "M is a model for S" as the assertion in the metatheory that for every formula phi in S there is a proof of that formula phi relativized to M from the axioms used at present. He says both definitions are of "different sorts". I suppose Kunen's point here is that no matter the theory at hand, there is a formula in the language of the formal theory asserting "phi is true in M", whereas there is a priori no formula in the language of the formal theory to assert that for all phi in S. It seems that if our theory can interpret some arithmetic (I do not know exactly how much) then there are formulas in the formal language that make assertions of the kind (b) (I suppose that at least for definable (recursive) sets of formulas). Please correct me if I am wrong. Then, my original question was if the word "model" in definition (b) has the same meaning as the word "model" for example in the sentence (c) "T is consistent iff T has a model".
From: MoeBlee on 16 Feb 2010 11:29 On Feb 16, 2:55 am, Marc Alcobé García <malc...(a)gmail.com> wrote: > my original question was if the word "model" in definition (b) > has the same meaning as the word "model" for example in the sentence > (c) "T is consistent iff T has a model". If I'm not mistaken, the answer is 'no'. If I recall, Kunen addresses that himself. Let's call the first notion 'class model' and the second notion 'set model'. If having a class model entailed consistency, then by using the class defining formula "x=x" we could prove the consistency of more theories than we would like to. Right? My understanding is that class models are for showing relative consistency; they don't by themselves prove plain consistency. Also, on a previous matter, I notice that Kunen, as I did, mentions specifically when he's talking about relative consistency or when he's talking about a theory proving the consistency of another theory - two distinct, though related notions. Also, on a previous matter, I notice that Kunen touches on handling of improperly referring terms and suggests what I call 'the Fregean method', though he doesn't call it by that name or explain it fully,. As to two earlier questions you had: (1) How to define 'meta-theory'. I don't know. I've tried but no luck. I started a thread about it about a year(?) ago. The other poster there thought it's not a good (or whatever his word) question. Anyway, to say that "M is a meta-theory for T" seems to be something that is said in meta-language for the languages of both M and T. One thing that M must do, at least, is define T, I would think. But I've had no luck formalizing the notion of defining a theory within a meta-theory. It seems (though it might not be) "ineffable", at least by my initial attempts. (2) You said you think relativization and interpretation within a theory are the same. But relativization occurs between a theory (or, if we wish to generalize, a set of formulas) and a formula. While interpretation of a theory within another theory occurs between two theories. So I think the notions are related but different. And you mentioned some other matters that I hope to address when I have more time. MoeBlee
From: Marc Alcobé García on 17 Feb 2010 03:19 > my original question was if the word "model" in definition (b) > has the same meaning as the word "model" for example in the sentence > (c) "T is consistent iff T has a model". I thought that both uses of the word "model" coincide when the M in definition (b) happens to be a set. But now I think this must be wrong because: 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...
From: Aatu Koskensilta on 17 Feb 2010 03:55
Marc Alcob� Garc�a <malcobe(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.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |