Abstract
In Chap. 1, we described what a first-order language is and what its terms and formulas are. We fixed a first-order language L. In Chap. 2, we described the semantics of first-order languages. In Chap. 3, we considered a simpler form of logic – propositional logic, defined what a proof is in that logic, and proved its completeness theorem. In this chapter we shall define proof in a first-order theory and prove the corresponding completeness theorem. The result for countable theories was first proved by Godel in 1930. The result in its complete generality was first observed by Malcev in 1936. The proof given below is due to Leo Henkin.
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