Let $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ be the following principle ("determinacy for simple open length-$\omega_1$ games"):

If $\kappa$ is any ordinal and $X\subseteq \kappa^{<\omega_1}$ is in $L(\mathbb{R})$, then the game of length $\omega_1$ on $\kappa$ with payoff set $$[X]:=\{f\in\kappa^{\omega_1}: \exists\alpha\in\omega_1(f\upharpoonright\alpha\not\in X)\}$$ is determined in $V$.

My question is whether $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ is consistent relative to large cardinals. I recall seeing a while ago a proof that the answer is yes, relative to something like a proper class of Woodins, but I can't track it down. (I'm pretty sure I can prove that this principle is equivalent to its clopen game analogue, but that's no easier to prove consistent as far as I can tell.) Neeman's work on long games is the most relevant I can find, but doesn't seem to directly address this unless I'm missing something.

  • $\begingroup$ I'm confused. Doesn't your principle imply $\text{AD}_{\omega_1}$ for games of length $\omega$ using countable ordinal moves? If so, it is inconsistent. That is, for $\kappa=\omega_1$, I seem to be able to mimic games on countable ordinals of length $\omega$ in your context. But there is a nondetermined such game. $\endgroup$ May 7 at 1:17
  • $\begingroup$ @JoelDavidHamkins Doesn't such a game have payoff set not in $L(\mathbb{R})$? $\endgroup$ May 7 at 1:19
  • $\begingroup$ $\text{AD}_{\omega_1}$ is inconsistent with ZF, which holds in $L(\mathbb{R})$. $\endgroup$ May 7 at 1:20
  • $\begingroup$ Ah, you mean the strategy might not be in $L(\mathbb{R})$. I see. $\endgroup$ May 7 at 1:20
  • $\begingroup$ @JoelDavidHamkins Yes, the game's payoff set needs to be in $L(\mathbb{R})$, but the strategy just has to live in $V$. $\endgroup$ May 7 at 1:21


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