Precipitous Ideal Separation at Higher Cardinals

Definitions and mathematical statement

Work in $\mathsf{ZFC}$. For an uncountable regular cardinal $\kappa$, use the following convention: an ideal on $\kappa$ is a proper family $I\subseteq\mathcal P(\kappa)$ containing every singleton, closed under subsets and unions of fewer than $\kappa$ members. Thus it contains every bounded subset of $\kappa$. The supplied extract does not explicitly specify this completeness convention; the questions below use it and require confirmation against the original report.

Write $I^+=\mathcal P(\kappa)\setminus I$. Forcing with $\mathcal P(\kappa)/I$ means forcing with the nonzero equivalence classes modulo $I$, ordered by $[A]_I\leq[B]_I$ if $A\setminus B\in I$. If $G$ is generic over the ground universe $V$, let \[ U_G=\{A\in\mathcal P(\kappa)^V:[A]_I\in G\}. \] The ideal $I$ is precipitous if this forcing forces the generic ultrapower $\operatorname{Ult}(V,U_G)$, formed using ground-model functions, to be well-founded. The ideal is normal if, for every sequence $\langle A_\alpha:\alpha<\kappa\rangle$ of members of $I$, its diagonal union satisfies \[ \mathop{\bigtriangledown}_{\alpha<\kappa}A_\alpha =\{\beta<\kappa:\exists\alpha<\beta\ (\beta\in A_\alpha)\}\in I. \] A cardinal $\delta$ is supercompact if, for every cardinal $\lambda\geq\delta$, there is an elementary embedding $j:V\to M$ into a transitive class $M$ such that $\operatorname{crit}(j)=\delta$, $j(\delta)>\lambda$, and ${}^{\lambda}M\subseteq M$, with closure computed in the ambient universe.

Consistency questions. Under these conventions, ask separately whether each of the following theories has a model:

1. $\mathsf{ZFC}$ together with \[ \begin{gathered} \text{there exists a precipitous ideal on }\aleph_2,\\ \text{there exists no normal precipitous ideal on }\aleph_2. \end{gathered} \] The second clause excludes every normal precipitous ideal on $\aleph_2$, rather than merely requiring the exhibited ideal to be nonnormal.
2. $\mathsf{ZFC}$ together with \[ \begin{gathered} \text{there exists a supercompact cardinal},\\ \text{there exists no precipitous ideal on }\aleph_1. \end{gathered} \]

All cardinals, ideals, and quantifiers in each question are interpreted in the proposed model. The broader request to generalize a separation result beyond $\aleph_1$ supplies motivation, but the extract does not state that antecedent result or a precise generalization for arbitrary higher cardinals. No particular relative-consistency hypothesis is supplied.

Short English statement

Ask whether a model of $\mathsf{ZFC}$ can have a precipitous ideal on $\aleph_2$ while having no normal precipitous ideal on that cardinal. Separately, ask whether a supercompact cardinal can coexist with the absence of precipitous ideals on $\aleph_1$. The ideal conventions and the scope of the broader higher-cardinal generalization require source review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30001680$, source problem code catalogue entry, title Precipitous Ideal Separation at Higher Cardinals. Collection: Oberwolfach Reports Open Problems; proposed year: 2011. No individual proposer is identified in the supplied record.
• Supplied report citation: Set Theory (2011), Oberwolfach Reports, pp. 85--140, DOI 10.4171/owr/2011/02. Source URL and literature-source label "Oberwolfach problem statement": https://doi.org/10.4171/owr/2011/02. The supplied record says the official report confirms the explicit higher-cardinal questions and assigns the statement status corrected_verified; these are supplied verification claims, not independent verification here.
• Supplied literature-source label: "Gitik, precipitous ideals without normality": https://doi.org/10.1142/S0219061312500080. The record supplies no fuller bibliographic details.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the cited $\aleph_1$ construction did not settle the higher-cardinal and supercompact variants, which it assesses as remaining open. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.