So I've been thinking about the axiom of choice (AC) and the axiom of determinacy (AD), which are known to be inconsistent with each other. Some qualitative similarities of them are:
- they are both trivial in the 'finite' case
- they have useful consequences (the AC arguably more so, but e.g. AD implies that all subsets of $\mathbb R$ are measurable, so for the sake of the argument lets say they are both useful)
Now I wonder: the axiom of choice has also seemingly paradoxical (e.g. Banach-Tarski) or seemingly 'too strong' implications (e.g. the well ordering theorem). Is there a similar example concerning the axiom of determinacy? I.e. is there an example of a consequence of AD which in some sense feels paradoxical or too strong?