Journal of Symbolic Logic

On Generic Extensions Without the Axiom of Choice

G. P. Monro

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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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J. Symbolic Logic, Volume 48, Issue 1 (1983), 39-52.

First available in Project Euclid: 6 July 2007

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Monro, G. P. On Generic Extensions Without the Axiom of Choice. J. Symbolic Logic 48 (1983), no. 1, 39--52.

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