Logic and Proof MT24, Vaught's test

Suppose:

• $\sigma$ is a countable signature
• $\pmb T$ is a $\sigma$-theory with no finite models (necessary so you can’t extend $\pmb T$ with a sentence which separates finite/infinite structures)
• Any two countable models of $\pmb T$ are isomorphic

Then:

• $\pmb T$ is complete.

Suppose that $\pmb T$ is not complete. Then there is some sentence $F$ such that both $\pmb T \cup \lbrace F\rbrace$ and $\pmb T \cup \lbrace \lnot F\rbrace$ are satisfiable. Since $\pmb T$ has no finite models, $T \cup \lbrace F \rbrace$ and $T \cup \lbrace \lnot F \rbrace$ must have infinite models.

By the compactness theorem, there exists a countable model $\mathcal A$ of $\pmb T \cup \lbrace F\rbrace$ and a countable model $\mathcal B$ of $\pmb T \cup \lbrace \lnot F\rbrace$.

But by assumption, $\mathcal A$ and $\mathcal B$ are isomorphic. This is a contradiction, since $\mathcal A \models F$ while $\mathcal B \models \lnot F$.

(The proof fails in the case where there might be finite models)