I've been going over the extremely interesting discussions about Axiom of Choice.


It looks to me like all the "weird" consequences of AC (Banach-Tarski etc) come from using it on uncountable collections of sets.


If, instead, we only believe the Axiom of Countable Choice, do we still get unintuitive consequences in the same sense ?


Apologies in advance if the question is vague.



Peculiar examples with Axiom of Countable Choice ?
If you assume the existence of suitable large cardinals, then $L(\mathbb{R})$ is a model of the Axiom of Determinacy $AD$ and the Axiom of Dependent Choice $DC$. In particular, since $DC$ is stronger than the Axiom of Countable Choice $AC\_{\omega}$, it follows that $AC\_{\omega}$ is also true in $L(\mathbb{R})$. Since $L(\mathbb{R})$ satisfies $AD$, all subsets $X \subseteq \mathbb{R}$ and maps $f: \mathbb{R} \to \mathbb{R}$ are measurable, etc. So it seems extremely unlikely that you will find any unintuitive consequences of $AC\_{\omega}$ in the more classical areas of mathematics.


