Baldwin-Lachlan Theorem
Theorem: (Baldwin
and Lachlan,
JSL 1971)
If T is an uncountably categorical theory in a countable language, then either T is
countably categorical as well or T has exactly ℵ0 countable models up to
isomorphism.
They show that if M is a model of an
uncountably categorical theory
(in a countable language) then there is
a strongly minimal set X definable in M
over parameters from the prime model and
that M is prime over X.
