Nearly Model Complete Theories

A theory T of a language L is 1‐model complete (nearly model complete) iff for every formula ρ of L there is a formula ϕ ( χ ) of L which is a ∀∃‐formula (a Boolean combination of universal formulas) such that T ⊨ ∀ x [ϕ↔θ]. The main results of the paper give characterizations of nearly model complete theories and of 1‐model complete theories. As a consequence we obtain that a theory T is nearly model complete iff whenever is a model of T and ⊆ 1 , then T ∪ Δ 1 is a complete L (A)‐theory, where Δ 1 is the 1‐diagram of . We also point out that our main results extend to ( n + l)‐model complete and nearly ra‐model complete theories for all n > 0.