Godel's Constructible Universe and Standard Models of ZFC

Question

I take the following to be a commonly-accepted definition of an inner model (in the context of ZFC):

Let (M,∈) be a model of ZFC. The pair (N,E) is an inner model of ZFC in M if:
(i) (N,E) is a model of ZFC
(ii) E is the restriction of ∈ to N
(iii) N is a transitive class of M
(iv) The class Ord is the same in N as it is in M.

If (N,E) satisfies just (i) and (ii), then we say that (N,E) is a standard model of ZFC in M. Sometimes we also say that (N,E) is a standard submodel of ZFC in (M,∈).

Now, as is explained here, in any given universe, the existence of a standard model of ZFC is stronger than simply the existence of a model of ZFC. This is also pointed out in the Wikipedia article on standard models (here). However, it is both stated later in that very same Wikipedia article and is mentioned in an article on Cantor's Attic (here) that Godel showed that any model of ZFC has a least inner model, called the constructible universe. This seems to give us that Con(ZFC) implies the existence of a model implies the existence of a standard submodel, since inner models are certainly standard submodels. This chain of implication is problematic insofar as it appears to contradict the fact that Con(ZFC) is strictly weaker than the existence of a standard model of ZFC.

I suspect that the issue at hand is caused by (a) sloppiness regarding just what universe our statements apply to in what situation, and (b) sloppiness regarding the conditions under which we assert Con(ZFC) in the first place. Here is what I see as the resolution: As is discussed by Asaf Karagila in the first link above, when we say that ZFC is consistent, we are doing so in some grand universe V of sets, which (together with some relation ∈) we typically take to be a model of ZFC. In the context of V, to say that ZFC is consistent is to say that there is some set M and some relation E on M, both in V, such that (M,E) is a model for ZFC. That is, Con(ZFC) tells us only about the existence of a certain object living in V with certain properties. L, on the other hand, is a proper class of our universe V, i.e is not a member of V. From the inside of V, then, L is not seen as existing; much less is it seen as being a standard model. From the outside of V, however, L looks like a standard submodel of V. But as soon as we're external to V, then our domain of discussion is some bigger universe V', which (together with some relation ∈') we take to be a model of ZFC. However, since ∈ need not be the restriction of ∈' to V, L is not necessarily a standard submodel of ZFC in V'.

That is, in order to know that something is an inner model of V, we have to know that it exists in the first place, and sometimes this requires that our domain of discussion be bigger than V. In the case of Godel's constructible universe L, we know that it is an inner (and hence standard) model of V only when our domain of discussion V' is bigger than V; and in this case, L need not be a standard model relative to our bigger universe V'.

Is this resolution plausible? Thank you in advance.

Answer 1

I’m not sure I understand everything you say in your resolution, but it captures the main ideas that we need to be careful about internal vs external, set vs proper class, and where things are proved.

The statement that is stronger than Con(ZFC) is that there is a transitive set model of ZFC. This statement, along with the weaker statement that a not-necessarily-transitive set model exists (which is equivalent to Con(ZFC)) are statements we can fully formalize as sentences in the language of set theory.

For proper class “models” like $V$ or $L$, on the other hand, we can’t necessarily internalize the idea that they are models, since there is generally no satisfaction relation for proper classes (certainly never for $V$ by Tarski’s theorem, though we can sometimes get one for $L$ under large cardinal assumptions). Instead, the idea that they are models must be understood as a scheme of relativized axioms proven one at s time. $V$ is quite trivially a transitive proper class model of ZFC by this standard, so we can see that the “existence” of a transitive proper class model doesn’t mean anything on its own.

What we get from the existence of proper class models are relative consistency results in the metatheory rather than consistency results in the theory. For instance the fact that every axiom of ZFC + V=L relativized to $L$ can be proved in ZF implies that the consistency of ZF implies the consistency of ZFC + V=L. This can be viewed as a purely proof theoretical result in a finitary metatheory, or as being about nested models, viewed externally as set models in a set theory.