## Notre Dame Journal of Formal Logic

### The Decidability of the $\forall^*\exists$ Class and the Axiom of Foundation

#### Abstract

We show that the Axiom of Foundation, as well as the Antifoundation Axiom AFA, plays a crucial role in determining the decidability of the following problem. Given a first-order theory T over the language $=,\in$, and a sentence F of the form $\forall x_1, \ldots, x_n \exists y F^M$ with $F^M$ quantifier-free in the same language, are there models of T in which F is true? Furthermore we show that the Extensionality Axiom is quite irrelevant in that respect.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 42, Number 1 (2001), 41-53.

Dates
First available in Project Euclid: 30 May 2003

https://projecteuclid.org/euclid.ndjfl/1054301354

Digital Object Identifier
doi:10.1305/ndjfl/1054301354

Mathematical Reviews number (MathSciNet)
MR1993389

Zentralblatt MATH identifier
1023.03007

#### Citation

Bellè, Dorella; Parlamento, Franco. The Decidability of the $\forall^*\exists$ Class and the Axiom of Foundation. Notre Dame J. Formal Logic 42 (2001), no. 1, 41--53. doi:10.1305/ndjfl/1054301354. https://projecteuclid.org/euclid.ndjfl/1054301354

#### References

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