# Are there seemingly paradoxical consequences of the axiom of determinacy?

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?

• The fact that all subsets of $\Bbb R$ are measurable seems too strong if not paradoxical. – Arnaud Mortier Nov 6 '19 at 17:51
• In my view this feels more like Tychonoffs theorem, i.e. that all products of compact spaces are compact, which seems too strong, but is just so useful. But of course that is a matter of opinion. – Redundant Aunt Nov 6 '19 at 18:18
• Well, even if you restrict Tychonoff to Hausdorff spaces, you still contradict AD. – Asaf Karagila Nov 6 '19 at 18:20