Recursive Equivalence

Abstract

Let L be a complete effectively presented countable first order language with identity. Let L(C) be the extension of L obtained by adding a fully effective countably infinite set of individual constants. If T′ is a complete theory in L(C) and T a theory in L then T′ is said to be a witness completion of T if T′ ⊇ T and whenever ∃ υ _n φ (υ _n ) is in T, then φ (c) is in T′ for some c ∈ C. The Henkin-Hasenjaeger proof of completeness of first order logic has an obvious effective version obtained by simply making all listings effective (see Mendelson [1964] exercise p. 65 and Proposition 2.12). There is also an obvious relativized version.