Let M be a short extender mouse. We prove that if $E\in M$ and $M\models $“E is a countably complete short extender whose support is a cardinal $\theta $ and $\mathcal {H}_\theta \subseteq \mathrm {Ult}(V,E)$”, then E is in the extender sequence $\mathbb {E}^M$ of M. We also prove other related facts, and use them to establish that if $\kappa $ is an uncountable cardinal of M and $\kappa ^{+M}$ exists in M then $(\mathcal {H}_{\kappa ^+})^M$ satisfies the Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then $\mathbb {E}^M$ is definable over the universe of M from the parameter $X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$, and M satisfies “Every set is $\mathrm {OD}_{\{X\}}$”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “$V=\mathrm {HOD}$”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters $u_n$.

A short core model induction proof of $\mathsf {AD}^{L(\mathbb {R})}$ from $\mathsf {TD} + \mathsf {DC}_{\mathbb {R}}$.

We construct divergent models of $\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis ($\mathsf {CH}$) under various assumptions. Divergent models of $\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in $\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of $\mathsf {AD}^+$. Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings of the reals and principles implying $\neg \mathsf {CH}$, like $\mathsf {MM}$, that divergent models shed light on, see Question 5.1.

We show that in many extender models, e.g., the minimal one with infinitely many Woodin cardinals or the minimal with a Woodin cardinal that is a limit of Woodin cardinals, there are no generic embeddings with critical point $\omega _1$ that resemble the stationary tower at the second Woodin cardinal. The meaning of “resemble” is made precise in the paper (see Definition 0.3).

We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n(x)[g]$ for a Turing cone of reals x, where $M_n(x)$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol \Pi ^1_{n+2}$ -determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n(x)[g]} = M_n(\mathcal {M}_{\infty } | \kappa _{\infty }, \Lambda ),$ where $\mathcal {M}_{\infty }$ is a direct limit of iterates of $M_{n+1}$ , $\delta _{\infty }$ is the least Woodin cardinal in $\mathcal {M}_{\infty }$ , $\kappa _{\infty }$ is the least inaccessible cardinal in $\mathcal {M}_{\infty }$ above $\delta _{\infty }$ , and $\Lambda $ is a partial iteration strategy for $\mathcal {M}_{\infty }$ . It will also be shown that under the same hypothesis $\operatorname {HOD}^{M_n(x)[g]} $ satisfies $\operatorname {GCH} $ .

A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The ${\sf Largest\ Suslin\ Axiom}$ ( ${\sf LSA}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf over}$ - ${\sf uB}$ be the statement that in all (set) generic extensions there is a model of $\sf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, $\sf {Sealing}$ is equiconsistent with $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that $\sf {Sealing}$ is weaker than the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of $\sf {Sealing}$ by Woodin. A variation of $\sf {Sealing}$ , called $\sf {Tower \ Sealing}$ , is also shown to be equiconsistent with $\sf {Sealing}$ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then $\sf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that $\sf {Sealing}$ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ is not equivalent to $\sf {Sealing}$ .

We determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.

We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails.

We analyze the models $L[T_{2n} ]$, where $T_{2n}$ is a tree on $\omega \times \kappa _{2n + 1}^1 $ projecting to a universal ${\rm{\Pi }}_{2n}^1 $ set of reals, for $n > 1$. Following Hjorth’s work on $L[T_2 ]$, we show that under ${\rm{Det}}\left( {{\rm{}}_{2n}^1 } \right)$, the models $L[T_{2n} ]$ are unique, that is they do not depend of the choice of the tree $T_{2n}$. This requires a generalization of the Kechris–Martin theorem to all pointclasses${\rm{\Pi }}_{2n + 1}^1$. We then characterize these models as constructible models relative to the direct limit of all countable nondropping iterates of${\cal M}_{2n + 1}^\# $. We then show that the GCH holds in $L[T_{2n} ]$, for every $n < \omega $, even though they are not extender models. This analysis localizes the HOD analysis of Steel and Woodin at the even levels of the projective hierarchy.

Let Msw denote the least iterable inner model with a strong cardinal above a Woodin cardinal. By [11], Msw has a fully iterable core model, ${K^{{M_{{\rm{sw}}}}}}$, and Msw is thus the least iterable extender model which has an iterable core model with a Woodin cardinal. In V, ${K^{{M_{{\rm{sw}}}}}}$ is an iterate of Msw via its iteration strategy Σ.

We here show that Msw has a bedrock which arises from ${K^{{M_{{\rm{sw}}}}}}$ by telling ${K^{{M_{{\rm{sw}}}}}}$ a specific fragment ${\rm{\bar{\Sigma }}}$ of its own iteration strategy, which in turn is a tail of Σ. Hence Msw is a generic extension of $L[{K^{{M_{{\rm{sw}}}}}},{\rm{\bar{\Sigma }}}]$, but the latter model is not a generic extension of any inner model properly contained in it.

These results generalize to models of the form Ms (x) for a cone of reals x, where Ms (x) denotes the least iterable inner model with a strong cardinal containing x. In particular, the least iterable inner model with a strong cardinal above two (or seven, or boundedly many) Woodin cardinals has a 2-small core model K with a Woodin cardinal and its bedrock is again of the form $L[K,{\rm{\bar{\Sigma }}}]$.

We show that the Mouse Set Conjecture for sets of reals is true in the minimal model of ADℝ + “Θ is regular”. As a consequence, we get that below ADℝ + “Θ is regular”, models of AD++¬ADℝ are hybrid mice over ℝ. Such a representation of models of AD+ is important in core model induction applications.

In this paper, we show that the failure of the unique branch hypothesis $\left( \text{UBH} \right)$ for tame trees implies that in some homogenous generic extension of $V$ there is a transitive model $M$ containing Ord $\cup \mathbb{R}$ such that $M\,\vDash \,\text{A}{{\text{D}}^{+}}\,+\,\Theta \,>\,{{\theta }_{0}}$. In particular, this implies the existence (in $V$) of a non-tame mouse. The results of this paper significantly extend J. R. Steel's earlier results for tame trees.

The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture (MSC). One particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large.

Let κℝ be the least ordinal κ such that Lκ (ℝ) is admissible. Let A = {x ϵ ℝ ∣ (∃α < κℝ) such that x is ordinal definable in Lα (ℝ)}. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the theory: ZFC − Replacement + “There exists ω Woodin cardinals which are cofinal in the ordinals.” T has consistency strength weaker than that of the theory ZFC + “There exists ω Woodin cardinals”, but stronger than that of the theory ZFC + “There exists n Woodin Cardinals”, for each n ϵ ω. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A = ℝ ∩ M. Since M is a mouse, we say that A is a mouse set. As an application, we use our characterization of A to give an inner-model-theoretic proof of a theorem of Martin which states that for all n, every real is in A.

The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. determinacy implies that for every thin equivalence relation there is a real, N, over which every equivalence class is generic—and hence there is a good (N#) wellordering of the equivalence classes. Analogous results are obtained for and quasilinear orderings and determinacy is shown to imply that every prewellorder has rank less than .