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Undecidability results for restricted universally quantified formulae of set theory. (English) Zbl 0797.03005

Summary: The problem of establishing whether there are sets satisfying a formula in the first order set theoretic language \({\mathcal L}_ \in\) based on \(=\), \(\in\), which involves only restricted quantifiers and has an equivalent \(\forall\exists\)-prenex form (\((\forall\exists)_ 0\)-formula), is neither decidable nor semidecidable. In fact, given any \(\omega\)-model of \(\text{ZF}-\text{FA}\), where FA denotes the Foundation Axiom, the set of existential closures of \((\forall\exists)_ 0\)-formulae true in the model is a productive set. Undecidability arises even when dealing with restricted universal quantifiers only, provided a predicate is-a-pair \((x)\), meaning that \(x\) is a pair of distinct sets, is added to \({\mathcal L}_ \in\). If satisfiability refers to \(\omega\)-models of \(\text{ZF}- \text{FA}\) in which a form of Boffa’s antifoundation axiom holds, then semidecidability fails as well; in fact, given any such model, the set of existential closures of formulae involving only restricted quantifiers and the predicate is-a-pair which are true in it, is a productive set. These results are all proved by making use of appropriate codings of Turing machine computations in the set theoretic language.

MSC:

03B25 Decidability of theories and sets of sentences
03E99 Set theory
03D35 Undecidability and degrees of sets of sentences
03D10 Turing machines and related notions
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