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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.)

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    By "sentence" $p$ you presumably mean an axiom scheme, not a sentence in the traditional sense.2012-02-27
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    No! That's what fascinating about it: you only need a *single* first-order sentence - if you restrict your attention to infinite structures only. (Another way of saying this is that there is some sentence p such that T is the theory of consequences of p together with the scheme of sentences $q_n$, where $q_n$ asserts that the domain has size at least $n$.)2012-02-27
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    My confusion is that if $T$ is complete, and has infinite models, and $p$ is true in some or all infinite models, then by completeness $p$ is true in all models.2012-02-27
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    @André: The important part is that $p$ must be false in all _non-models_. That does not follow from completeness alone.2012-02-28

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