The Disposition of Complete Theories

The purpose of this paper is to give a purely logical proof of a result of Mostowski [1937] concerning the complete theories of a calculus based on classical propositional logic; and then modestly to generalize it. Mostow-ski’s result is announced by Tarski on p. 370 of Logic, Semantics, Meta-mathematics [1956]. (All references to Tarski’s work here are to this book.) Tarski himself provides only a fragment of a proof, and the proof published by Mostowski makes extensive use of topological methods and results. The a proof offered here is undoubtedly longer than Mostowski’s and not by any means independent of it. But it should not be beyond the powers of any-one who has followed assiduously a couple of courses in propositional logic and knows a little set theory. The axiom of choice is assumed, but not the continuum hypothesis. A calculus is what Tarski called a deductive system (originally a deduc- b tive theory): the set of sentences of a language plus an operation Cn of logical consequence based on that of classical propositional logic. A theor