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On Generic Extensions Without the Axiom of Choice
G. P. Monro
The Journal of Symbolic Logic
Vol. 48, No. 1 (Mar., 1983), pp. 39-52
Published by: Association for Symbolic Logic
Stable URL: http://www.jstor.org/stable/2273318
Page Count: 14
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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[ G] of ZF (a "generic extension"). If the axiom of choice holds in M it also holds in M[ G], 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.
The Journal of Symbolic Logic © 1983 Association for Symbolic Logic