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The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined.

In "Turing determinacy and the continuum hypothesis" (published in 1989), Ramez Sami writes:

"The main question so far unsettled in this particular domain can be roughly put this way: is it true that for any "reasonable" pointclass $\Gamma$ we have: Turing-Det$(\Gamma)\implies$Det$(\Gamma)$? In particular is it the case that: [over $ZF+DC$, presumably] Turing $AD$ implies $AD$?"

My question is, what is the status of this question currently? Do we know whether Turing $AD$ is strictly weaker than $AD$? The only recent work I know of around Turing determinacy is from the reverse mathematical side (; I'm not at all familiar with the set theory on the subject.

(I vaguely recall that Turing determinacy implies that every Suslin set is determined, but I can't remember where I supposedly learned this "fact.")

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up vote 8 down vote accepted

This is open. In $L(\mathbb R)$ the answer is yes. Hugh has several proofs of this, and it remains one of the few unpublished results in the area. The latest version of the statement (that I know of) is the claim in your parenthetical remark at the end. This gives determinacy in $L(\mathbb R)$ using, for example, a reflection argument.

(I mentioned this a while ago somewhere on this site. Maybe that's where you heard of it? This can be used to prove that $\omega$-board determinacy is equiconsistent with determinacy. I seem to recall that's how the topic came up.)

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Based on your answer, I googled around and found this:…. So that must be what I was remembering. It's probably too much to ask, but do you know if there is a version of the proof floating around? – Noah Schweber Jun 16 '14 at 2:00
I don't think so, unfortunately. Hugh lectured on this at the seminar at Berkeley a few years ago, but I couldn't find anybody who had taken notes. – Andrés Caicedo Jun 16 '14 at 2:01
(And, yes, you are right, it was on MSE that this came up.) – Andrés Caicedo Jun 16 '14 at 2:02

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