Stationary Reflection and Large Power at Aleph Omega

Definitions and mathematical statement

Work in $\mathsf{ZFC}$. For an infinite cardinal $\kappa$, write $\kappa^+$ and $\kappa^{++}$ for its first and second successor cardinals. Say that $\kappa$ is strong limit if $2^\lambda<\kappa$ for every cardinal $\lambda<\kappa$, and singular if $\operatorname{cf}(\kappa)<\kappa$. Here $2^\lambda$ denotes the cardinality of the power set of $\lambda$.

For a limit ordinal $\delta$, a set $C\subseteq\delta$ is club if it is unbounded in $\delta$ and closed under limits below $\delta$. A set $S\subseteq\delta$ is stationary in $\delta$ if it meets every club subset of $\delta$. Under the convention used here, stationary sets $S_0,\ldots,S_{n-1}\subseteq\theta$, where $\theta$ is regular uncountable, reflect simultaneously if there exists $\alpha<\theta$ with $\operatorname{cf}(\alpha)>\omega$ such that each $S_i\cap\alpha$ is stationary in $\alpha$.

Using this convention, simultaneous finite stationary-set reflection at $\theta$ means \[ \begin{gathered} \text{for every integer }n\geq 1\text{ and every sequence }\langle S_i:i<n\rangle \text{ of stationary subsets of }\theta,\\ \text{there exists }\alpha<\theta\text{ with }\operatorname{cf}(\alpha)>\omega \text{ such that }S_i\cap\alpha\text{ is stationary in }\alpha \text{ for every }i<n. \end{gathered} \] Consistency question. Can the consistency result referred to in the supplied record be obtained with $\kappa=\aleph_\omega$? Under the convention above, its requested target is a model of $\mathsf{ZFC}$ in which \[ \kappa=\aleph_\omega,\qquad \operatorname{cf}(\kappa)=\omega,\qquad (\forall\lambda<\kappa\text{ a cardinal})\;2^\lambda<\kappa, \qquad 2^\kappa=\kappa^{++}, \] and simultaneous finite stationary-set reflection holds at $\kappa^+=\aleph_{\omega+1}$.

Unresolved qualifications. The supplied record does not define its reflection principle. The displayed formulation uses all stationary subsets of $\kappa^+$ and requires a common reflection point of uncountable cofinality; whether the intended principle restricts the stationary sets or the cofinality of the reflection point must be checked against the source. The record calls the referenced result a $\Sigma$-Prikry consistency result, but supplies neither its exact theorem nor its large-cardinal hypotheses. Consequently, this draft does not specify a relative consistency strength or assert that a particular forcing construction is required.

Short English statement

The question asks whether $\aleph_\omega$ can be a strong-limit singular cardinal whose power set has size $\aleph_{\omega+2}$, while every finite family of stationary subsets of $\aleph_{\omega+1}$ reflects at a common ordinal of uncountable cofinality. This formulation uses an explicit reflection convention whose agreement with the intended source principle requires review. The supplied record labels the problem open; that status is not independently verified here.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30004447$, source problem code catalogue entry, title: Stationary Reflection and Large Power at Aleph Omega. The registry identifies the collection as Oberwolfach Reports Open Problems and supplies proposed year 2020; no individual proposer is identified.
• Supplied primary citation: Set Theory (2021), Oberwolfach Reports, pp. 797--855, DOI 10.4171/owr/2020/14. https://doi.org/10.4171/owr/2020/14. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. Its literature-source label is Oberwolfach Report 14/2020.
• Supplied literature source: Bilinsky--Gitik (2025). https://link.springer.com/article/10.1007/s00153-025-00983-4. No article title or specific supporting passage is supplied.
• Supplied literature assessment, checked on 2026-08-22: the specific $\Sigma$-Prikry consistency result was stated as open for $\kappa=\aleph_\omega$; related later literature on ultrafilters and singular-cardinal constructions was assessed as not establishing this exact simultaneous consistency result. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record expressly calls for independent verification of time-sensitive status and cited claims.