On the Role of the Baire Category Theorem and Dependent Choice in the Foundations of Logic

Abstract

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 Rasiowa-Sikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces.