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.
Citation
Dorella Bellè. Franco Parlamento. "Decidability and Completeness for Open Formulas of Membership Theories." Notre Dame J. Formal Logic 36 (2) 304 - 318, Spring 1995. https://doi.org/10.1305/ndjfl/1040248461
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