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The following is a theorem by Vaught.

Theorem. Let $T$ be a complete theory in a countable language. Then, $T$ cannot have exactly two countably infinite models (up to isomorphism).

A proof can be found at [Tent-Ziegler, A Course in Model Theory, Theorem 4.3.10].

Does the theorem still hold in an uncountable language? That is:

Question. Let $T$ be a complete theory in an uncountable language. Can $T$ have exactly two countably infinite models?

Any help would be appreciated. Thank you.

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