Complete theory with a infinite model has only infinite models

Question

Let $T$ be a complete first order theory with a infinite model $M$. I want to show that every model $N$ of $T$ must be infinite.

Since $T$ is complete then for every sentence $\phi$ we can either derive $\phi$ or $\neg \phi$ from $T$.

By a corollary of the Upwards Löwenheim-Skolem theorem, if $T$ is a consistent theory which has an infinite model of cardinality $\kappa, \kappa \geq \vert L \vert$. In particular if $L$ is countable then $T$ has models of every infinite cardinal.

So we know that $T$ has a lot of infinite models but how can I be sure that no finite structures model $T$. How can I show this?

Answer 1

Hint: suppose $M \models T$ is finite of size $n$. Can you come up with a first order statement that expresses that $M$ has size $n$? Then since $T$ is complete, that statement or its negation would be in $T$.

Answer 2

Any complete theory with a finite model is categorical. So if $T$ a complete theory has an infinite model, it is not categorical, so it cannot have a finite model.