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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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##### References:
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