The Non-Axiomatizability of O-Minimality

Fix a language L extending the language of real closed fields by at least one new predicate or function symbol. Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L-sentences \Lambda, there is a real closed field R satisfying \Lambda, which is not pseudo-o-minimal. In particular, there are locally o-minimal, definably complete real closed fields which are not pseudo-o-minimal. This answers negatively a question raised by Schoutens, and shows that the theory consisting of those L-sentences true in all o-minimal L-structures, called the theory of o-minimality (for L), is not recursively axiomatizable.