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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
DOI: 10.2307/2273318
Stable URL: http://www.jstor.org/stable/2273318
Page Count: 14
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On Generic Extensions Without the Axiom of Choice
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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[ 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.

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