Hybrid Inner Models at Limits of Woodin Cardinals

Definitions and mathematical statement

Let $V$ denote the ambient set-theoretic universe. For an infinite cardinal $\theta$, write \[ H_\theta^V=\{x\in V:|\operatorname{tc}(\{x\})|^V<\theta\}, \] where $\operatorname{tc}$ denotes transitive closure. Write $\operatorname{Ord}$ for the ordinals and $\mathrm{AD}$ for the axiom of determinacy: every length-$\omega$, perfect-information game in which the players alternately choose natural numbers is determined. The terms measurable, Woodin, and strong refer to their standard large-cardinal meanings. Interpret "$\kappa$ is a limit of Woodin and strong cardinals" as requiring both classes separately to be unbounded below $\kappa$: \[ \forall\alpha<\kappa\ \exists\delta,\eta<\kappa\, \left(\alpha<\delta\land\alpha<\eta\land \delta\text{ is Woodin}\land\eta\text{ is strong}\right). \] Here $\operatorname{Coll}(\omega,{<}\kappa)$ denotes the usual Lévy collapse of the ordinals below $\kappa$ to countable size.

The supplied conjecture assumes that there is no inner model with a Woodin cardinal which is a limit of Woodin cardinals. It asks whether every measurable $\kappa$ which is a limit of Woodin and strong cardinals admits hybrid structures \[ M_0\mathrel{\triangleleft}M_1\subseteq H_{\kappa^+} \] such that, for every generic $g\subseteq\operatorname{Coll}(\omega,{<}\kappa)$, all four requirements hold:

1. $H^-\mathrel{\triangleleft}M_0$.
2. $L(M_0^\omega,\Gamma,\mathbb R)\models\mathrm{AD}$.
3. $\operatorname{Ord}\cap M_0$ is the largest cardinal of $M_1$.
4. $M_1$ is close to $V$ at $\kappa$.

The existential choice of $M_0,M_1$ precedes the universal quantifier over generic filters, as in the supplied cleaned statement.

Unresolved qualifications. This is a statement schema requiring further source definitions. The record does not define hybrid structures, their language or additional predicates, the relation $\triangleleft$, $H^-$, the pointclass or parameter $\Gamma$, or the condition "close to $V$ at $\kappa$." Consequently these expressions are retained without substituting a particular technical interpretation. The record also does not specify the universe in which each conclusion is evaluated, or how $H^-$, $\Gamma$, $\mathbb R$, and $M_0^\omega$ depend on $g$. Conventionally one would read $\kappa^+$ and $H_{\kappa^+}$ in the ground universe, take $g$ to be generic over $V$, and evaluate the generic-dependent objects in $V[g]$; the supplied record does not establish this interpretation. Likewise, the construction denoted by $L(M_0^\omega,\Gamma,\mathbb R)$ needs its intended parameter or predicate convention. The largest-cardinal clause requires an appropriate well-founded structural interpretation in which $\operatorname{Ord}\cap M_0$ is an ordinal recognized by $M_1$. No missing structural hypotheses or definitions are asserted here.

Short English statement

Assuming there is no inner model with a Woodin cardinal that is itself a limit of Woodin cardinals, the conjecture asks for two nested hybrid structures below $H_{\kappa^+}$ whenever $\kappa$ is measurable and a limit of both Woodin and strong cardinals. The same structures must satisfy the four displayed requirements for every indicated collapse generic. Several essential structures, parameters, and the meaning of closeness are undefined in the supplied record, so a fully precise formulation requires further source context. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30003414$, title "Hybrid Inner Models at Limits of Woodin Cardinals," collection "Oberwolfach Reports Open Problems," exact source problem code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Set Theory (2018), Oberwolfach Reports, pp. 527--589, DOI 10.4171/owr/2017/11, https://doi.org/10.4171/owr/2017/11. The supplied background attributes the problem to open-problem discussions documented in Oberwolfach workshop reports. No problem-specific page or speaker is supplied.
• Supplied literature source, "OWR Set Theory report": https://ems.press/content/serial-article-files/46674.
• Supplied literature source, "Descriptive inner model theory research statement": https://sites.math.unt.edu/~ntrang/ResearchStmt.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the hybrid-structure covering assertion as conjectural, with later generation-of-pointclasses work providing partial results but not the exact four-part conclusion. This is supplied dated curation, not an independently verified literature assessment; the supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that notation, typography, and sentence structure were repaired while preserving mathematical content. These are supplied editorial claims, not mathematical verification of this draft or its unresolved terminology.