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.
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© 1974 Springer-Verlag Berlin Heidelberg
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Crossley, J.N., Nerode, A. (1974). Recursive Equivalence. In: Combinatorial Functors. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 81. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85933-5_5
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DOI: https://doi.org/10.1007/978-3-642-85933-5_5
Publisher Name: Springer, Berlin, Heidelberg
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