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?
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$\begingroup$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 $T$.
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$It also works if $T$ is not necessarily complete.
Let's show that if $T$ an $\mathcal{L}$-theory has an infinite model then $T$ has infinite models of arbitrary large cardinal (i.e for all set $X$ there exists a model $\mathfrak{M}\models T$ such that $\vert M\vert \ge \vert X\vert$ where $M$ is the domain of $\mathfrak{M}$).
Let's add to $\mathcal{L}$ the following set of new constants $\{c_x : x\in X\}$. Then let's define $\mathcal{L}'=\mathcal{L}\cup \{c_x : x \in X\}$ where the $(c_x)_{x\in X}$ are pairwise distincts.
Let's consider $T' = T \cup \{c_x \neq c_y : x,y \in X ;x\neq y \}$ and prove that $T'$ is consistent. Indeed if $\mathfrak{N}' \models T'$ (in particular $\mathfrak{N}'\models T$) hence $\mathfrak{N}:= \mathfrak{N}'_{\vert \mathcal{L}} \models T$ and if we denote by $N$ the common domain of $\mathfrak{N}$ and $\mathfrak{N}'$ then the map $f : X \to N,\ x \mapsto c_x^{\mathfrak{N}'}$ is injective.
Let $T_0\subseteq T'$ finite. There exists $X_0$ a finite set such that $T_0 \subset T \cup \{c_x \neq c_y : x,y \in X_0 ;x\neq y \}$. Let $\mathfrak{M}$ an infinite model of $T$ (the existence is provided by the hypothesis), $g$ an injective map from $X_0$ to $M$ and $a \in M$. Then we define $\mathfrak{M}'$ an expansion of $\mathfrak{M}$ to $\mathcal{L}'$ which is a model of $T \cup \{c_x \neq c_y : x,y \in X_0 ;x\neq y \}$ and in particular of $T_0$ by setting $\mathfrak{M}'_{\vert \mathcal{L}}=\mathfrak{M}$, for all $x\in X_0, \ g(x)=:c_x^{\mathfrak{M}'}$ and for all $x\in X\setminus X_0$, $c_x^{\mathfrak{M}'}:=a$. Then $T'$ is finitely consistent hence consistent by compactness theorem.
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