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Decidability and completeness for open formulas of membership theories. (English) Zbl 0837.03007
Summary: We establish 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 result of adding \(y\) as an element to \(x\), of the theory NW having the obvious axioms for \(\emptyset\) and \(w\). Furthermore we establish completeness with respect to purely universal sentences of the theory NW+E+R obtained from NW by adding the Extensionality Axiom E and the Regularity Axiom R, and of the theory \(\text{NW+AFA}'\) obtained by adding to NW (a slight variant of) the Antifoundation Axiom AFA.

MSC:
03B25 Decidability of theories and sets of sentences
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