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and look for his papers with the word cohomology in the title. I do not know if there is a GAGA for o-minimal cohomology, but at least in stupid cases (e.g. nonstandard expansions of the real numbers) if done correctly coholomogy with constant coefficients is independent of the choice of the expansion (even though topologically these nonstandard fields are pretty disconnected).
@Colin and Kevin: There is certainly a cohomology theory for semialgebraic varieties -- in fact, there is at least the beginning of a sheaf cohomology theory for any o-minimal structure. If you do a google search "cohomology semialgebraic varieties" you see get some interesting links. For cohomology theory in the o-minimal setting see the homepage of Mario Edmundo ciul.ul.pt/~edmundo
Just read the blog post. Incidentally, this question came up in Davesh's colloquium talk yesterday. He and I both thought that the formal group argument that Jared suggests is a good natural way to do this, except that formal groups are not exactly elementary, and that there are no formal groups for general algebraic dynamical systems (the latter remark is due to A. Medvedev).
@Jordan: The p-adic integration idea that Nathan and I use would count orders on the nose (a copy is available on my homepage). I don't know; maybe obtaining a mass formula is the right thing to do, but I don't know how to do that using our methods. AV and I were at some point trying to count (some particular) orders using the trace formula. We never completed the project, but in that case we would have obtained a mass formula involving regulators and sizes of automorphism groups.
@Esperantist: Thank you! This is awesome. As of yesterday I suspected that most subrings should be non-split, but I wasn't able to construct an example. This is very nice.
