Understanding the hypotheses in Morley's categoricity theorem.
I'm curious what kind of theories are ruled out by the no finite models hypothesis in Morley's theorem and if there's a good or canonical example of one that necessitates this hypothesis.
In this comment on this answer to this question I asked yesterday, I learned that Morley's categoricity theorem requires that the theory in question be complete, consistent, have no finite models, and have a countable language in order to be $\lambda$-categorical for all $\lambda > \omega$.
On page 286, A Shorter Model Theory says the following about constraints placed on $T$.
Throughout this section, $L$ will be a countable first-order language and $T$ a complete theory in $L$ with infinite models.
This passage is sort of ambiguous on its own, since the last part could mean that $T$ has at least one infinite model or it could mean that $T$ has at least one infinite model and no finite models. I looked at the definition of Morley's theorem briefly in the past, just to try to understand the statement of the theorem but not its proof, and I read it wrong the first time around.
So, now I'm sort of curious why that hypothesis is there. I'm especially curious if there's an example that's relatively easy to understand since I don't understand the proof of the theorem as a whole.
This is really just a nontriviality condition.
Only incomplete theories can have both finite and infinite models, since "There are exactly $n$ many elements in the domain" is expressible by a first-order sentence for each finite $n$. A complete theory which has a finite model in fact has a unique model up to isomorphism (while by contrast if $T$ has an infinite model then $T$ has a proper class of models up to isomorphism, due to compactness).
So in the context of complete theories and counting infinite models, having a finite model trivializes everything.