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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?

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    $\begingroup$ 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. $\endgroup$ – Max Sep 26 at 14:36
  • $\begingroup$ @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? $\endgroup$ – Alex Kruckman Sep 26 at 16:34
  • $\begingroup$ @AlexKruckman : you are of course right - I must misremember... but I don't think it was that question you linked to $\endgroup$ – Max Sep 26 at 17:33
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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.

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