Is the Axiom of Choice restricted to $\Bbb R$ independent of ZF?

Asked Jul 15 '14 at 06:14

Active Jul 15 '14 at 06:14

Viewed 39 times

This is really a yes-no question, and I am hundred percent certain that the answer is "yes". I simply have not found it written directly anywhere.

True, the Axiom of Choice is independent of ZF if we do not consider AC a part of ZF. But the question is: Let us restrict AC to $\Bbb R$ and just claim that whenever I have a subset of $\mathscr P(\Bbb R)\setminus\{\emptyset\}$, this subset has a choice function. Is this statement AC$\big|_{\Bbb R}$ independent of ZF?

asked Jul 15 '14 at 06:14

Gaussler

2,766

Yes.${}{}{}{}{}$ – Andrés E. Caicedo Jul 15 '14 at 06:17
Do you know a source where I can find a proof? :-) – Gaussler Jul 15 '14 at 06:22
Jech Set Theory Theorem 14.36 proves that there is a model in which the real numbers can not be well-ordered. – William Jul 15 '14 at 06:22

Is the Axiom of Choice restricted to $\Bbb R$ independent of ZF?

0 Answers0