## Journal of Symbolic Logic

### On Generic Extensions Without the Axiom of Choice

G. P. Monro

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183741188

Mathematical Reviews number (MathSciNet)
MR693246

Zentralblatt MATH identifier
0522.03034

JSTOR