Definability of Types, and Pairs of O-Minimal Structures

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 ⅼ M ⅼ + -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 * ⊢ (∀ x )( P ( x ) → ( ϕ ( x ) ↔ ψ ( 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 X = { X ⊂ 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 }.