Full disclosure (in order to forestall indignation): I'm going to be using this result in my upcoming paper (I have a statement which I show to be equivalent to $\neg CC(\mathbb R)$ and which I want to prove to be consistent).
So if you really don't want me to finish my paper, don't answer this question!
The axiom of countable choice for subsets of the real numbers states that any countable product of subsets of the reals is nonempty.
I need the proof of the statement that the negation of this axiom is consistent with ZF (or ZFA).
I'd be interested in both
- a readable proof of that proposition, as well as
- the information who proved it first.
Thank you very much in advance (especially if you can stomach my silly remarks).
Cohen constructed his first model of the failure of choice where there is an infinite Dedekind-finite set of reals.
Since every infinite set of reals can be mapped into $\omega$, such a surjection from our Dedekind-finite set will produce a countable family of non-empty sets, indeed a countable partition of the Dedekind-finite set. Since any choice function from this family will define an injection from $\omega$ into the set, there is no such choice function.