## Journal of Symbolic Logic

### Truth in V for ∃*∀∀-sentences is decidable

#### Abstract

Let V be the cumulative set theoretic hierarchy, generated from the empty set by taking powers at successor stages and unions at limit stages and, following [2], let the primitive language of set theory be the first order language which contains binary symbols for equality and membership only. Despite the existence of ∀∀-formulae in the primitive language, with two free variables, which are satisfiable in V but not by finite sets ([5]), and therefore of ∃∃∀∀ sentences of the same language, which are undecidable in ZFC without the Axiom of Infinity, truth in V for ∃*∀∀-sentences of the primitive language, is decidable ([1]). Completeness of ZF with respect to such sentences follows.

#### Article information

Source
J. Symbolic Logic, Volume 71, Issue 4 (2006), 1200-1222.

Dates
First available in Project Euclid: 20 November 2006

https://projecteuclid.org/euclid.jsl/1164060452

Digital Object Identifier
doi:10.2178/jsl/1164060452

Mathematical Reviews number (MathSciNet)
MR2275856

Zentralblatt MATH identifier
1109.03057

Subjects
Primary: Primary 03B25, Secondary 03E30, 03C62

#### Citation

Bellé, D.; Parlamento, F. Truth in V for ∃ * ∀∀-sentences is decidable. J. Symbolic Logic 71 (2006), no. 4, 1200--1222. doi:10.2178/jsl/1164060452. https://projecteuclid.org/euclid.jsl/1164060452

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