Decidability and Completeness for Open Formulas of Membership Theories

Abstract

We establish the decidability, with respect to open formulas in the first order language with equality =, the membership relation $\in$, the constant $\emptyset$ for the empty set, and a binary operation w which, applied to any two sets x and y, yields the results of adding y as an element to x, of the theory NW having the obvious axioms for $\emptyset$ and w. Furthermore we establish the completeness with respect to purely universal sentences of the theory $\mbox{NW}+\mbox{E}+\mbox{R}$, obtained from NW by adding the Extensionality Axiom E and the Regularity Axiom R, and of the theory $\mbox{NW}+\mbox{AFA}'$ obtained by adding to NW (a slight variant of) the Antifoundation Axiom AFA.