Finite models are atomic I want to show that any finite model is atomic. To prove this, it is enough to show that any type realized in a finite model is principal (isolated).
Let $T$ be a theory, $\mathcal{A}$ a finite model of $T$, and $p$ a type realized in $\mathcal{A}$. How can we show that $p$ is principal?
If $T$ is assumed complete, then we can show that $T$ is absolutely categorical, and so $p$ is realized in every model of $T$ and thus is isolated. But how would we show this without assuming $T$ complete? Thanks

A (complete) $n$-type is just a complete theory over the extended language where you adjoin $n$ new constant symbols, and a model together with a realization of the type is just a model of the complete theory in the extended language.  So it suffices to show that any complete theory (over a finite language) which has a finite model is generated by a single axiom.  This is easy: just write an axiom that completely describes your finite model up to isomorphism.
(As Alex Kruckman's answer shows, you do need to assume the language is finite.)
