# What sets can be unraveled?

A set $$X\subseteq\omega^\omega$$ is unravelable iff there is a possibly larger set $$A$$ and a clopen set $$Y\subseteq A^\omega$$ (with respect to the product topology coming from the discrete topology on $$A$$) such that winning strategies for the game on $$A$$ with payoff set $$Y$$ can be converted to winning strategies for the game on $$\omega$$ with payoff set $$X$$ in a particularly simple way (see Martin's paper A purely inductive proof of Borel determinacy for the precise definition). By Gale–Stewart, unravelability implies determinacy; however, it is not the only route to proving the determinacy of a pointclass.

My question is whether there is a known upper bound on the complexity of an unravelable pointclass. To make this somewhat precise:

Is it consistent with $$\mathsf{ZFC}$$ that every set in $$\mathcal{P}(\mathbb{R})^{L(\mathbb{R})}$$ is unravelable?

The strongest results I can find are vastly weaker than this, namely that under large cardinal assumptions $$\Pi^1_1$$ sets are unravelable (Neeman 1,2; I remember seeing the same result for Borel-on-$$\Pi^1_1$$ sets, but I can't find a reference for it at the moment). On the other hand, I don't see an easy proof that this is anywhere close to the most unravelability we can expect, nor can I find this stated in the literature.

## 1 Answer

I emailed Itay Neeman, and he told me the following:

As far as I know it's open. I don't think anything is known about unraveling beyond what you can get from my methods. These give the Suslin operation on $$\Pi^1_1$$ sets, and various iterations of that. In terms of the large cardinal hierarchy it's still all just using measures. Anything above that is open I think.

In particular I don't think it's known if $$\Sigma^1_2$$ sets can be unraveled.

For now I think that settles this question. (Itay later said that it's unclear whether we "should" expect unravelability of $$\Sigma^1_2$$ sets from large cardinals or not in the first place. So I think my main takeaway at this point is that unravelability is not best thought of as yet another tameness property, where the slogan "large cardinals make definable things tame" is expected to prevail.)

• The opposite of settled, but that makes it more interesting, no? Mar 10, 2022 at 4:05
• @AndrésE.Caicedo Indeed. I'm curious: do you have a sense as to whether unravelability of $\Sigma^1_2$ or more is something we should expect from large cardinals? (I asked Itay, and his response was that it was unclear at present but I'm curious if anyone else has ideas.) Mar 11, 2022 at 17:54