Models of set theory vs. models of first order theories
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From: MoeBlee on 8 Feb 2010 13:15 On Feb 6, 2:08 am, Marc Alcobé García <malc...(a)gmail.com> wrote: > But to prove Con(T) you already need the consistency of something > else... don't you? Perhaps you have in mind the epistemological sense of proof as an argument that convinces of a conclusion beyond doubt (or some similar sense)? That is not what I am talking about. I mean 'proof' in the formal sense of a formal deduction in formal system with axioms (0 or more) and rules of inference. For a theory S to prove 'Con(T)', it is not required that S be consistent. All that is required is that 'Con(T)' be in the language of theory S and that 'Con(T)' is derivable, per S, by the rules of inference from the axioms. For example, Z set theory proves the consistency of first order PA. This is the case since in Z we can express the statement that PA is consistent, and prove, in Z, the statement that PA is consistent. And if Z is not consistent, then as long as Z can still express the statement that PA is consistent, then Z proves that statement anyway. The consistency of the theory S that proves the conistency of another theory T is not required just for us to see that theory S does prove the consistency of T (that is, the consistency of T is proven not necessarily in some epistemological sense, but rather at least in the formal sense). So back to the difference between a consistency theorem and a relative consistency theorem: Let's write "P is provable in theory S" as: |-_S P Now, say P is some consistency statement about a theory T. Then |-_S P or, |-_S Con(T) says theory S proves the consistency of theory T (REGARDLESS whether theory S is itself consistent). On the other hand, let's say P states the consistency of theory T and Q states the consistency of theory T*. Then |-_S (P -> Q) or |-_S (Con(T) -> Con(T*)) says that theory S proves that T* is consistent RELATIVE to theory T. For example, in ZF we can prove: Con(ZF) -> Con(ZFC) I.e. |-_ZF (Con(ZF) -> Con(ZFC)) To recap: |-_S Con(T) says that S proves a consistency statement, viz. the consistency of T. |-_S (Con(T) -> Con(T*)) says that S proves a relative consistency statement, viz. the consistency of T* relative to the consistency of T. MoeBlee
From: MoeBlee on 8 Feb 2010 13:37 On Feb 8, 3:08 am, Marc Alcobé García <malc...(a)gmail.com> wrote: > Clearly there are two uses of the word "model". One is to refer to > structures for first-order languages which make true a set of > formulas > in a given language (the nonlogical axioms of a theory). I might state it a little differently, but okay, what you said is good enough. > Structures > are anything but sets. I don't know why you say 'structures are anything but sets'. > The other one is to refer to relative > interpretations of a theory T in another theory T'. In this last case > all we can say is that if T' has a model, then T has a model, thereby > obtaining only a relative consistency proof. I think that is correct. > Now, the question can be rewritten as: is any statement of the form > Con(T) a statement in a given language of a certain theory T' (the > metatheory) which interprets T (the theory)? Then, our proof of > Con(T) > would be only a proof of Con(T') -> Con(T). Maybe there are authors who say that, but it's not how I would say it. Rather, if T* proves Con(T), then I would write: |-_T* Con(T). That is different from a relative consistency, which would be written: |-_T* (Con(T) -> Con(S)) I.e., T* proves that if T is consistent then S is consistent. For example: |-_ZF (Con(ZF) -> Con(ZFC)) which is of a different form from |-_ZF Con(PA). So, if, when T* proves the consistency of T, you write: Con(T*) -> Con(T) you leave out what theory THAT statement is proven in. If instead, you write: |-_T* Con(T) then that is precise and clear. Moreover, even if T* is consistent, while |-_T* Con(T) it could still be that T* has false axioms (finitary factually false) allowing that it could be false (finitary factually false) that T is consistent. That's another reason writing Con(T*) -> Con(T) may be misleafing (it might be thought of as stating some fact, in such form) where all we really have is |-_T* Con(T) while it could still be that T* is consistent but T is inconsistent (due to T* having a false axiom). > there is no way we > can prove [the] consistency [of elementary arithmetic] unless we > use a stronger metatheory that > elementary arithmetic cannot interpret. (Perhaps depending on what you mean by 'elementary arithmetic') I already pointed out that the consistency proving theory may be a stronger theory OR a theory that is NOT COMPARABLE as to strength of the object theory. Why do you keep skipping that? MoeBlee
From: Marc Alcobé García on 9 Feb 2010 15:47 > Perhaps you have in mind the epistemological sense of proof as an > argument that convinces of a conclusion beyond doubt (or some similar > sense)? That is not what I am talking about. You have hit the nail on the head. In Kunen's words I try to think about the metatheory as "what is really true" (p. 7). But this idea does not seem to fit very well with seeing the metatheory as a theory interpreted in (an extension by definitions of) ZFC.
From: MoeBlee on 9 Feb 2010 16:00 On Feb 9, 2:47 pm, Marc Alcobé García <malc...(a)gmail.com> wrote: > > Perhaps you have in mind the epistemological sense of proof as an > > argument that convinces of a conclusion beyond doubt (or some similar > > sense)? That is not what I am talking about. > > You have hit the nail on the head. In Kunen's words I try to think > about the metatheory as "what is really true" (p. 7). That's okay, if we make sure that our meta-theory uses only finitistic matheamtics, otherwise, what do you consider to be "really true"? (And, I do understand that certain relative consistency results can be proven finitistically.) But still, we may regard the meta-theory as formalized (at least in principle), and that formalization is usually itself a ZFC (or something like that). And that does not contradict that certain results may also be proven (at least in principle) in weaker finitistic systems. In any case, we're still talking about consistency of (object) theories, which is a formal notion. No matter what we think of our meta-theory being "really true" or whatever, it still is good to be clear about the distinction between consistency and relative consistency. > But this idea > does not seem to fit very well with seeing the metatheory as a > theory > interpreted in (an extension by definitions of) ZFC. I wouldn't say the meta-theory is "interpreted" (which is a techncial term) but rather that the meta-theory simply IS (a copy of; whatever) ZFC (or whatever your favorite meta-theory). (I'm not saying that one can't carry out interpretation in the technical sense in the way you mention regarding meta-theories; only I'm saying that I don't see an immediate reason for complicating the matter in that way.) MoeBlee
From: MoeBlee on 10 Feb 2010 11:05
On Feb 8, 12:37 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > That is different from a relative consistency I thought of another reason it helps to keep the distinction between, on the one hand, a theory proving the consistency of another theory, and on the other hand, relative consistency. I'll use an example: ZF+"there exists an inaccessible cardinal" (I'll call it 'ZFI') proves the consistency of ZF. So I would write that as: (1) |-_ZFI Con(ZF). But, as a relative consistency statement, it would be: (2) Con(ZFI) -> Con(ZF). But (2) doesn't inform me very much, since its proof is trivial - a trivial consequence of the monotonicity of the logic, while (1) is informative to me, since it leads me to look at its proof, which is showing that V_i (where i is some inaccessbible cardinal) is a universe for a model of ZF. Meanwhile, the relative consistency statement Con(ZF) -> Con(ZFC) is informative (the non-trivial proof due to Godel) but it is not the same as |-_ZF Con(ZFC) which is not even true. So there is an important distinction between (1) a theory proving the consistency of another theory and (2) a relative consistency. MoeBlee |