## O-minimality and uniform finiteness

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. The first big question about o-minimality is the following: is o-minimality an elementary property, that is, given ${\cal N} \equiv {\cal M}$, is ${\cal N}$ necessarily o-minimal? Lemma The following are equivalent: every ${\cal N} \equiv {\cal M}$ is o-minimal; for every…