Notes - 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 $\pmb T \cup \{F\}$ and $\pmb T \cup \{\lnot F\}$ are consistent.

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

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