On generic extensions without the axiom of choice

Journal of Symbolic Logic 48 (1):39-52 (1983)

Abstract

Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let $M$ be a countable transitive model of ZF. The method of forcing extends $M$ to another model $M\lbrack G\rbrack$ of ZF (a "generic extension"). If the axiom of choice holds in $M$ it also holds in $M\lbrack G\rbrack$, that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive an application to Boolean toposes

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References found in this work

Principia mathematica.A. N. Whitehead & B. Russell - 1910-1913 - Revue de Métaphysique et de Morale 19 (2):19-19.
Boolean-Valued Models and Independence Proofs in Set Theory.J. L. Bell & Dana Scott - 1986 - Journal of Symbolic Logic 51 (4):1076-1077.
Topos Theory.P. T. Johnstone - 1982 - Journal of Symbolic Logic 47 (2):448-450.

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