A variant of the proof of the completeness of the first order functional calculus

Abstract

It will be proved that every true formula of the first order functional cal? culus is valid in it. The first proof of this theorem known as the theorem of comple? teness of the functional calculus of first order was given by Kurt G?del in 1930. The present proof1 follows those involving the concept of prime ideal, especially the proof given by Juliusz Reichbach in [I]2. The characteristic feature of the present proof is that the general theory of deductive systems constructed by A. Tarski in [3] is essential for it, and espe? cially Lindenbaum's theorem on complete supersystems. The proof is similar to the proof of completeness of sentential calculus presented in [2]. 1. We assume that the letters X, Y, Z, ... in the axioms of Tarski's theory designate subsets of the set of all meaningful formulae, and the letters a, ?, y,... the elements of this set. This convention will be observed in further remarks on Tarski's theory. Al. ?< Ko A2. X C CnX c 5 A3. If XcY, then CnX C CnY A4. CnCnX C CnX