Indecomposable Ultrafilters Beyond Measurable Cardinals

Definitions and mathematical statement

The supplied question is: "Can there be an indecomposable ultrafilter on a cardinal that is neither measurable nor the limit of countably many measurable cardinals?" The record does not define "indecomposable" or specify uniformity, a lower bound on the cardinal, or the intended consistency assumptions. Consequently, the following is a conditional formalization requiring confirmation of these conventions.

Let $\kappa$ be an uncountable cardinal. An ultrafilter on $\kappa$ is a family $U\subseteq\mathcal P(\kappa)$ such that $\kappa\in U$, $\varnothing\notin U$, $U$ is closed under finite intersections and supersets within $\kappa$, and exactly one of $A$ and $\kappa\setminus A$ belongs to $U$ for every $A\subseteq\kappa$. It is uniform if $|A|=\kappa$ for every $A\in U$.

One convention for indecomposability of a uniform ultrafilter $U$ on $\kappa$ is \[ \forall\text{ cardinals }\lambda<\kappa\quad \forall f:\kappa\longrightarrow\lambda\quad \exists A\in U\quad |f[A]|\leq\aleph_0. \tag{*} \] Equivalently, every partition of $\kappa$ into fewer than $\kappa$ pieces has a countable subfamily whose union belongs to $U$. The supplied record does not confirm that $(*)$ is its intended definition.

An uncountable cardinal $\mu$ is measurable if it carries a nonprincipal $\mu$-complete ultrafilter: intersections of any family of fewer than $\mu$ members of that ultrafilter remain in the ultrafilter. Interpret "the limit of countably many measurable cardinals" as the existence of a strictly increasing sequence $\langle\mu_n:n<\omega\rangle$ of measurable cardinals below $\kappa$ such that $\sup_{n<\omega}\mu_n=\kappa$.

Under these conventions, the candidate mathematical question is whether there can be a model of $\mathrm{ZFC}$ containing an uncountable cardinal $\kappa$ and a uniform ultrafilter $U$ on $\kappa$ satisfying $(*)$, such that \[ \kappa\text{ is not measurable} \quad\text{and}\quad \neg\exists\langle\mu_n:n<\omega\rangle\, \left[ \begin{gathered} \forall n<\omega\; (\mu_n\text{ is measurable and }\mu_n<\mu_{n+1}<\kappa),\\ \sup_{n<\omega}\mu_n=\kappa \end{gathered} \right]. \] All cardinal and ultrafilter properties here are interpreted inside the model in question. This consistency reading, the uncountability restriction, and the uniformity requirement are provisional interpretations rather than hypotheses explicitly present in the source question. No particular relative-consistency hypothesis or consistency strength is supplied.

Short English statement

The question asks whether an indecomposable ultrafilter can occur at a cardinal outside the measurable cardinals and their countable increasing limits. A candidate precise formulation is given using uniform ultrafilters on uncountable cardinals, but the source does not specify these conventions or the consistency assumptions. The supplied status is partially_solved; it is a dated curation label, not an independently verified mathematical conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004445$, source problem code catalogue entry, title "Indecomposable Ultrafilters Beyond Measurable Cardinals." The record labels the statement exact and says it was retained byte-for-byte. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied primary report 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 records of mathematical workshops and their open-problem discussions. The same report is also labeled "Oberwolfach Report 14/2020" in the supplied literature list.
• Supplied literature reference: Sheard (1983), https://doi.org/10.2307/2273664.
• Supplied literature reference: Goldberg (2020), https://arxiv.org/abs/2007.04812.
• Supplied literature reference: Oberwolfach Report 14/2020, https://doi.org/10.4171/owr/2020/14.
• Literature assessment dated exactly 2026-08-22: this is supplied dated curation, not independently verified. It reports that Sheard showed consistency of an indecomposable ultrafilter on an inaccessible non-weakly-compact cardinal; that canonical inner models give the opposite behavior; and that, above a strongly compact cardinal, Goldberg proved the measurable-or-countable-limit classification. The supplied assessment characterizes the unrestricted question as independence or relative-consistency territory rather than a $\mathrm{ZFC}$ theorem and labels its status partially solved. The background explicitly calls this a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.