Definability of types, and pairs of o-minimal structures

Journal of Symbolic Logic 59 (4):1400-1409 (1994)
Let T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L * be L together with a unary predicate P. Let T * be the L * -theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an |M| + -saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T * is complete and we give a simple set of axioms for T * . We also show that for every L * -formula φ(x) there is an L-formula ψ(x) such that $T^\ast \models (\forall \mathbf{x})(P(\mathbf{x}) \rightarrow (\phi(\mathbf{x}) \mapsto \psi (\mathbf{x}))$ . This yields the following result: Let M be a Dedekind complete model of T. Let φ(x, y) be an L-formula where l(y) = k. Let $\mathbf{X} = \{X \subset M^k$ : for some a in an elementary extension N of M, X = φ (a,y N ∩ M k }. Then there is a formula ψ(y, z) of L such that X = {ψ (y, b) M : b in M}
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