  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.
