TITLE: Characterization of totally categorical theories
QUESTION [5 upvotes]: I have what I am sure is a trivial question, but I can't seem to answer it for myself.
In model theory, there is a theorem of Hrushovski which shows that if T is a totally categorical theory (i.e., T is complete and has exactly one model of each infinite cardinality up to isomorphism), then (i) T is not finitely axiomatizable, but (ii) T is finitely axiomatizable modulo infinity; that is, there is some sentence p such that T is precisely the set of sentences true in every infinite model of p.
My question is to what extent the converse holds. That is, let T be a (EDIT: complete) theory which is finitely axiomatizable modulo infinity, but which is not finitely axiomatizable. Then is T necessarily totally categorical, and if not, what sort of assumptions on T are enough to ensure total categoricity?
(The assumption that T is not actually finitely axiomatizable is clearly necessary: otherwise, take the theory DLO of dense linear orders without endpoints, which is countable categorical but not uncountable categorical.)

REPLY [3 votes]: This is a nice question, it's too bad it was unanswered for so long. I'll give an example of a complete theory $T$ which is quasi-finitely axiomatizable (i.e. finitely axiomatizable modulo infinity) and not finitely axiomatizable, but not categorical in any cardinality.
Let $L = \{<\}$, and consider the theory $T$ of infinite discrete linear orders with endpoints. This says:
(1) $<$ is a linear order.
(2) There is a maximum element.
(3) There is a mininum element.
(4) If $x$ is not the maximum element, $x$ has a successor.
(5) If $x$ is not the minimum element, $x$ has a predecessor.
(6$_n$)$_{n\in\omega}$ There are at least $n$ elements.
Showing that $T$ is complete is a standard application of Ehrenfeucht-Fraïssé games.
Since (1)-(5) can each be expressed by a single sentence, the conjunction of these gives a finite axiomatization of $T$ modulo infinity: (6$_n$)$_{n\in\omega}$.
However, $T$ cannot be finitely axiomatizable, because it is pseudofinite. To spell this out, note that (1)-(5) are satisfied in any finite linear order. So if $\Delta$ is any finite collection of sentences of $T$, by compactness $\Delta$ is implied by a finite collection $\Delta'$ of the axioms of $T$ listed above. Now $\Delta'$ is satisfiable in any sufficiently large finite linear order, and hence so is $\Delta$. But $\Delta$ also has infinite models, since $T$ does, so $\Delta$ does not axiomatize a complete theory.
To finish, note that $T$ is not uncountably categorical (it's unstable), nor is it countably categorical (the formulas $\phi_n(x,y): \exists^{!n}z\, (x<z \land z < y)$, expressing that the distance between $x$ and $y$ is $(n+1)$, are distinct for all $n\in\omega$).