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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  exercise p. 65 and Proposition 2.12). There is also an obvious relativized version.
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