Limits of determinacy on reals

For $X\subseteq\mathbb{R}^\omega$, say that $X$ is determined if the associated game on $\mathbb{R}$ of length $\omega$ (players I and II alternate playing reals, player I wins iff the sequence built is in $X$) is determined.

My question is: what is known about the consistency of determinacy principles for games played with reals? For example, is it consistent with large cardinals that every $X\subseteq\mathbb{R}^\omega$ is determined? (I suspect not, but I'm having trouble coming up with a counterexample.)

EDIT: This actually splits into two questions; I'm interested in each:

How much determinacy on $\mathbb{R}$ is consistent (relative to large cardinals) with ZF?

Andreas' answer completely settles this question.

However, I'm also interested in:

How much determinacy on $\mathbb{R}$ is consistent (relative to large cardinals) with ZFC?

• Noah, I am not sure I understand your last question. How do you measure "how much"? Do you expect a precise boundary (As in: This pointclass is consistently determined, and that pointclass, that in such and such technical sense is the "next one" is not) or rather something more informal? Jan 20, 2015 at 1:27
• @AndresCaicedo, obviously I'd love a precise boundary, but I don't imagine such a thing is known. I was hoping that the state-of-the-art might be reasonably concise, though - that maybe there's a pointclass which is consistently determined, and no pointclass "much more complicated" is known to be consistently determined. I'd also be interested in the dual question - do we know a pointclass which is not consistently determined, such that no "much simpler" pointclass is known to be not consistently determined? That's still informal, but hopefully clearer - is that better? Jan 20, 2015 at 1:39

3 Answers

$\text{AD}_{\mathbb{R}}$ is equiconsistent with the existence of a $\lambda$ which is a limit of Woodin cardinals and cardinals which are $< \lambda$-strong. This large cardinal hypothesis is also known as the $\text{AD}_{\mathbb{R}}$-hypothesis. In other words this is also the same as having a strategic model with $\omega$ many Woodin cardinals.

• Hi! More precisely, a limit of Woodin cardinals and cardinals that are strong up to $\lambda$ (that is, they are strong in $V_\lambda$ rather than strong in $V$). Jan 20, 2015 at 1:17
• Hi Andres. You are indeed right, the $\kappa$'s have to be strong up to $\lambda$. Let me add this in the edit. Jan 20, 2015 at 1:20
• What is a "strategic model?" Jan 20, 2015 at 1:21
• Hi @Noah. The term refers to the hybrid mice that appear in the analysis of $\mathsf{HOD}$ (hybrid meaning that they are not purely fine structural but keep track of the relevant strategies along their construction). Perhaps the most accessible reference is Grigor's very nice paper on the BSL. Jan 20, 2015 at 1:24
• A link to Grigor Sargsyan's paper, "Descriptive Inner Model Theory": projecteuclid.org/download/pdf_1/euclid.bsl/1368716862 Jan 20, 2015 at 2:08

The statement that every subset of $\mathbb R^\omega$ is determined is called $AD_{\mathbb R}$, and it's consistent relative to large cardinals. I don't remember exactly how large, but I vaguely recall that it's only a little beyond what's needed for AD. (Surely an expert will soon stop by and provide the exact answer.)

• Perhaps surprising is that determinacy for arbitrary games on $\aleph_1$ is inconsistent with ZF. I believe this goes back to Mycielski's first paper on AD. Jan 19, 2015 at 22:32
• I realized I was unclear about the base theory I'm interested in; I've edited the question to be clearer. Also, re: determinacy on $\aleph_1$, see Andreas' answer to mathoverflow.net/questions/100196/indeterminacy-of-long-games. Jan 19, 2015 at 22:36

I know this is an old question, but here's a little information about the $\textsf{ZFC}$ case to anyone finding this question.

In my MO question, Juan kindly pointed out that Woodin has a result stating that it's consistent relative to a sharp for a Woodin limit of Woodins that every $\textsf{OD}(\mathbb R)$ game of length $\omega_1$ on the reals is determined. This result can be found in Neeman's book on long games, exercise 7F.15.

Beyond that, we can find a non-determined definable game of length $\omega_1+\omega$ on the reals (equivalently the integers). After $\omega_1$ many rounds, player I has played a sequence $X$ of reals. If $X$ contains a perfect subset then we can define a well-ordering of the reals via $X$ and thus a non-determined set of reals $A\subseteq \omega^\omega$. Player I is then supposed to spend his last $\omega$ moves to land in $A$, making the game non-determined. If $X$ did not contain a perfect subset then the last $\omega$ rounds is spent playing the perfect set game on $X$, which is non-determined as $X$ doesn't have the perfect set property. It seems like this game is $\Delta^2_2$, so that's at least a lower definability bound for inconsistency. When we get to length $\mathfrak c+\omega$ then we only have to consider the first case above, so that the game is (at most) $\Pi^1_2$ (for all strategies there's a move which makes the strategy non-winning).

I'm not sure what happens below these lower bounds, except that Borel determinacy at least ensures that all Borel games are determined, no matter the length (just play elements of ${^\alpha}\omega$ for $\omega$ many rounds instead of playing elements of $\omega$ for $\alpha\omega$ many rounds). Here's an attempt to illustrate the situation, where red indicates inconsistencies and blue indicates consistency relative to large cardinals. I'm not sure if more is known.