On the role of the Baire Category Theorem and Dependent Choice in the foundations of logic

The Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Cech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the "Henkin method" of proving deductive completeness of logical systems. The RasiowaSikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces. The relevance of the Baire Category Theorem to the fundamental metalogical principle of deductive completeness has long been known. Rasiowa and Sikorski [1950], in their Boolean-algebraic proof of Gddel's completeness theorem for firstorder logic, applied the Baire Category Theorem to the compact Hausdorff Stone space of a Boolean algebra to obtain their celebrated lemma about the existence of ultrafilters respecting countably many meets. Grzegorczyk, Mostowski, and RyllNardzewski [1961] later adapted this approach to obtain the completeness theorem for co-logic, by applying the Baire theorem to the complete metric space of co-models of an w-complete theory. A similar argument may also be developed for omittingtypes theorems. The aim of this paper is to isolate that part of the Rasiowa-Sikorski Lemma that does not depend on the Ultrafilter Theorem. A result about the existence of certain filters is obtained that is dubbed "Tarski's Lemma" since it is closely allied to Tarski's algebraic proof of the Rasiowa-Sikorski Lemma, as reported by Feferman [1952]. It will be shown that in set theory without choice, Tarski's Lemma is equivalent to each of (i) the Baire Category Theorem for Cech-complete spaces, (ii) the Baire Category Theorem for complete metric spaces, (iii) the Principle of Dependent Choice, (iv) the existence theorem for generic sets of forcing conditions, and (v) a proof-theoretic principle which abstracts the technique introduced by Henkin [1949] for proving completeness proofs. From this follows a proof that the Rasiowa-Sikorski Lemma is equivalent to the conjunction of the Ultrafilter Theorem with the Baire Category Theorem for compact Hausdorff spaces. Throughout this paper, assertions that one statement implies another, or that two statements are equivalent (imply each other) will mean that the implications Received November 1, 1983; revised February 23, 1984. ? 1985, Association for Symbolic Logic 0022-48 12/85/5002-0016/$02. 1 0