Nearly Model Complete Theories

Abstract

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[ϕθ]. 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 𝔅 is a model of T and 𝔄⊆1𝔅, then T ∪ Δ1𝔄 is a complete L(A)-theory, where Δ1𝔄 is the 1-diagram of 𝔄. We also point out that our main results extend to (n + l)-model complete and nearly ra-model complete theories for all n > 0.