Say two players ($A$ and $B$) are playing a game of length $\omega$ on $\omega_1$ in which they play a countable ordinal turn by turn (starting with $A$) and form a sequence of countable ordinals, $(a_0,b_0,a_1,b_1, \ldots)$ (they can see each other’s moves) of length $\omega$. Player $B$ wins if the set of the (values of the) obtained sequence is an ordinal, $A$ wins otherwise. Can player $B$ have a winning strategy in ZF + “$\omega_1$ is regular”(by which I mean, “Can ZF + ”$\omega_1$ is regular” prove that there is a winning strategy for $B$?”)?
Obvious strategy is to play every ordinal under every ordinal played by the opponent, which requires choice of some form, and I am not seeing how to do this using only the fact that $\omega_1$ is regular.
Note that converse is trivially true so if the answer is “yes”, then we have an equivalence. Thank you.