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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 |

### Keywords:

undecidability in first-order set theoretic language; ZF without Foundation Axiom; codings of Turing machine computations in set theoretic language; satisfiability; \(\omega\)-models; antifoundation axiom
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\textit{F. Parlamento} and \textit{A. Policriti}, Commun. Pure Appl. Math. 46, No. 1, 57--73 (1993; Zbl 0797.03005)

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### References:

[1] | Non-well-founded sets, CSLI Lecture Notes, 1988. |

[2] | Breban, Adv. Appl. Math. 5 pp 147– (1984) |

[3] | Breban, Comm. Pure Appl. Math. 34 pp 177– (1981) |

[4] | , and , Set-theoretic reductions of Hilbert’s tenth problem, pp. 65–75 in: Proceedings Third Workshop ”Computer Science Logic,” Kaiserslautern, 1989, · Zbl 0925.03045 |

[5] | , and , eds., Lecture Notes in Computer Science, Vol. 440, Springer-Verlag, Heidelberg, 1990. |

[6] | Cantone, Comm. Pure Appl. Math. 40 pp 37– (1987) |

[7] | and , Computability, Complexity, and Languages, Academic Press, New York, 1983. |

[8] | Parlamento, Comm. Pure Appl. Math 41 pp 221– (1988) |

[9] | Parlamento, Zeitscrift Math. Log. Grundlagen Math. 38 (1992) |

[10] | On a generalization of Herbrand’s theorem, Doctoral Dissertation, New York University, 1990. |

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