Is there any meaningful sense in which we can talk about o-minimal sentences of $L_{\omega_1,\omega}$? I can give a first attempt, easily; given a countable fragment $F$ and a sentence $\Phi$ in that fragment (or a conjunction of elements of that fragment, so $\Phi$ is an $F$-theory), say $\Phi$ is *o-minimal* if:

- $\Phi$ makes $<$ a linear order, and
- For every $\phi(x,\overline y)$ in $F$ and every finite tuple $\overline a$, the set $\phi(x,\overline a)$ is a finite union of points and open intervals.

Amusingly the conditions above are actually a single sentence (although not typically in $F$) so some questions about elementarity may become vacuous.

My real question is: has this every been studied? Are the usual theorems (cell decomposition, etc.) stateable and proveable in this framework? Are there any interesting examples?

If the answer to the first question is no, I'd really like to know why not. Is there some reason we should expect this not to be interesting? Or is it just the the o-minimal people and the infinitary logic people aren't the same people?

first orderdefinable sets, so is o-minimal in one sense you might define it. In fact this is o-minimal wrt the fragment generated by the formula "for all $x$, there is an $n\in\mathbb Z$ where $-n<x<n$." So this fits, but I'm not sure how interesting the example is. $\endgroup$ – Richard Rast May 29 '15 at 1:56