The question is as in the title, but let me explain a bit.

Assuming a proper class of Woodin cardinals, $L(\mathbb{R})$ satisfies AD (and DC). And $L(\mathbb{R})$ is a very natural inner model. I'm curious if there is a similarly natural inner model for AD$_\mathbb{R}$.

Now, ZF + $V=L(\mathcal{P}(\mathbb{R}))$ + large cardinals proves AD$_\mathbb{R}$ if I recall correctly; however, this is somewhat misleading as $L(\mathcal{P}(\mathbb{R}))$ is never a model of AD, let alone AD$_\mathbb{R}$, since in it the reals are well-ordered.

So I'm curious: assuming large cardinals, is there a reasonably canonical inner model of AD$_\mathbb{R}$ which is "easy to describe"?

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    $\begingroup$ Sure, there is a minimal pointclass $\Gamma$ such that $L(\Gamma,\mathbb R)$ satisfies $\mathsf{ZF}+V=L(\mathcal P(\mathbb R))+\mathsf{AD}_{\mathbb R}$. Minimality is here in the sense of Wadge degrees. The same applies to just about any natural determinacy assumption we can obtain using derived models of fine-structural inner models. I assume Grigor's BSL paper and his paper in the Memoirs of the AMS are the natural place to find details. $\endgroup$ – Andrés E. Caicedo May 7 '17 at 21:53
  • $\begingroup$ (Well, OK, as long as the assumption does not yet put us in the region of "divergent" $\mathsf {AD}$ models.) $\endgroup$ – Andrés E. Caicedo May 7 '17 at 21:58
  • $\begingroup$ @AndrésE.Caicedo Ah, thanks very much! I'm not very familiar with inner model theory (and I'm totally derived-model-illiterate) so that hadn't occurred to me. If you post this as an answer, I'll definitely upvote it - and I'll accept it as soon as I'm able to tease the necessary details from the papers you link (or others that I find). $\endgroup$ – Noah Schweber May 7 '17 at 22:01
  • $\begingroup$ Thank you. Let's wait a couple of days and if no answers show up in the meantime I will try to write something semicoherent as an answer. $\endgroup$ – Andrés E. Caicedo May 7 '17 at 22:14
  • $\begingroup$ @Andrés I think it's okay even if there are divergent models of AD: any minimal model of $\mathsf{ZF} + \mathsf{AD}_\mathbb{R}$ containing all reals is in fact minimum by the fact that the intersection of divergent models of AD containing all reals satisfies $\mathsf{AD}_\mathbb{R}$. $\endgroup$ – Trevor Wilson May 7 '17 at 22:57

A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\in\Gamma$. Note that if $\Gamma\subseteq\mathcal P(\mathbb R)$ and $L(\Gamma,\mathbb R)\models \Gamma=\mathcal P(\mathbb R)$, then $\Gamma$ is a Wadge initial segment. The relevance of this notion is that if $M$ is an inner model containing all the reals and satisfying $\mathsf{AD}_{\mathbb R}$, then $\Gamma=\mathcal P(\mathbb R)^M$ is a Wadge initial segment and $L(\Gamma,\mathbb R)\models\mathsf{AD}_{\mathbb R}$.

Under appropriate large cardinal assumptions, there is a Wadge initial segment $\Gamma=\Gamma_{min}$ such that $L(\Gamma,\mathbb R)\models\mathsf{AD^+}+\mathsf{AD_{\mathbb R}}+\Gamma=\mathcal P(\mathbb R)$. Moreover, given any inner model $M$ containing all the reals and satisfying $\mathsf{AD}^++\mathsf{AD}_{\mathbb R}$, we have $\Gamma_{min}\subset M$. The mention of $\mathsf{AD}^+$ may well be superfluous here (or, perhaps, we should redefine $\mathsf{AD}_{\mathbb R}$ as $\mathsf{AD^+}+\mathsf{AD_{\mathbb R}}$); the situation does not seem entirely understood otherwise.

Surely $\Gamma_{min}$ admits a purely descriptive set-theoretic description as well (in terms of the complexity of the iteration strategies of the hybrid or hod mice that it captures), but I do not know how to specify it.

I suspect all of this is written up in reasonable detail nowadays. I suggest to read first

MR3362806 Reviewed. Sargsyan, Grigor. Hod mice and the mouse set conjecture (English summary), Mem. Amer. Math. Soc. 236 (2015), no. 1111, viii+172 pp. ISBN: 978-1-4704-1692-8

(with all the technical details of the underlying theory) and

MR3087400 Reviewed. Sargsyan, Grigor(1-RTG). Descriptive inner model theory (English summary), Bull. Symbolic Logic 19 (2013), no. 1, 1–55

(for a more leisurely introduction).

The result can of course be generalized to other strengthenings of $\mathsf{AD}^+$, but you will eventually run into difficulties, as it is possible that there are incompatible (or ``divergent'') $\mathsf{AD}^+$ models, that is, it is consistent to have sets of reals $A,B$ such that $L(A,\mathbb R)$ and $L(B,\mathbb R)$ are both models of $\mathsf{AD}^+$, but $A$ and $B$ are Wadge-incomparable so $A\notin L(B,\mathbb R)$, $B\notin L(A,\mathbb R)$, and $L(A,B,\mathbb R)$ is not a model of $\mathsf{AD}^+$. In such a setting, it may well be that there is no minimal pointclass $\Gamma$ playing the role of $\Gamma_{min}$ for your strengthening of determinacy. What saves us for $\mathsf{AD}^++\mathsf{AD}_{\mathbb R}$ is that it is a theorem of Woodin that if $A,B$ are as above, and $\Gamma=\mathcal P(\mathbb R)^{L(A,\mathbb R)}\cap\mathcal P(\mathbb R)^{L(B,\mathbb R)}$, then ($\Gamma$ is again a Wadge initial segment, and) $L(\Gamma,\mathbb R)\models\mathsf{AD}^++\mathsf{AD}_{\mathbb R}$.

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    $\begingroup$ Do you happen to know what kind of large cardinal assumptions are necessary/sufficient for existence of such an initial segment? $\endgroup$ – Wojowu Oct 5 '17 at 15:08
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    $\begingroup$ A Woodin limit of Woodin cardinals is more than enough. It wouldn't surprise me if the optimal assumption is the existence of a sufficiently iterable $M_{adr}^\sharp$ (so, just a bit beyond the large cardinal assumption needed to establish the consistency of $\mathsf{AD}_{\mathbb R}$). $\endgroup$ – Andrés E. Caicedo Oct 5 '17 at 15:25

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