# Los Vaughts Test, why do we need out theory to be $\kappa$ Categorical

Im currently studying for my model theory exam and I came across Los Vaughts test. It states

Let $$\mathcal{L}$$ be countable and $$\Sigma$$ a $$\mathcal{L}$$-theory. $$\Sigma$$ is complete, if

1) $$\Sigma$$ only has infinite Models

2) There is a cardinal number $$\kappa$$ for which two models of cardinality $$\kappa$$ are isomorphic.

I get that all models have to be infinite to use Löwenheim Skolem. But in the proof we use the isomorphism just to get elementary equivalence.

Let $$A,B \in Mod (\Sigma)$$, then $$A \equiv A' \cong B' \equiv B$$ hence $$A \equiv B$$.

So wouldn't it be enough to just ask for elementary equivalence in the second statement?

• You are correct, it would be enough. But in practice, the notion of $\kappa$-categoricity comes more often. I think there was a question on this site a while ago about whether your condition implied $\kappa$-categoricity. – Max Sep 26 at 14:36
• @Max The question raised in your comment wouldn't make much sense: Any complete theory with infinite models satisfies the property that all models of size $\kappa$ are elementarily equiavlent (for every infinite $\kappa$), and of course this doesn't imply $\kappa$-categoricity. Is it possible you were thinking of this question? – Alex Kruckman Sep 26 at 16:34
• @AlexKruckman : you are of course right - I must misremember... but I don't think it was that question you linked to – Max Sep 26 at 17:33

Yes, you are correct. We can replace "all models of size $$\kappa$$ are isomorphic" with "all models of size $$\kappa$$ are elementarily equivalent" and everything goes through fine.