Measurability Assumption in the Prikry Extension Theorem

Definitions and mathematical statement

Work in a ground model $V$ of $\mathrm{ZFC}$. The generalized continuum hypothesis (GCH) is the assertion that $2^\mu=\mu^+$ for every infinite cardinal $\mu$. A cardinal $\kappa$ is strongly inaccessible if it is uncountable, regular, and satisfies $2^\mu<\kappa$ for every cardinal $\mu<\kappa$.

Write \[ [\kappa^+]^{<\kappa}=\{x\subseteq\kappa^+:|x|<\kappa\}. \] A subset of this set is club if it is unbounded under inclusion and closed under unions of increasing chains of nonzero length less than $\kappa$. A subset is stationary if it meets every such club. For a set of ordinals $x$, let $\operatorname{otp}(x)$ denote its order type in the inherited ordering. Define \[ \mathrm{SC}(\kappa)\quad\Longleftrightarrow\quad \kappa\text{ is strongly inaccessible and } S_\kappa=\{x\in[\kappa^+]^{<\kappa}:\operatorname{otp}(x)\text{ is a cardinal}\} \text{ is stationary}. \] Here a cardinal is identified with its initial ordinal, and these definitions are evaluated in $V$.

For an infinite cardinal $\lambda$, use the following convention for $\square_{\lambda,<\omega}$: there is a sequence \[ \langle\mathcal C_\alpha:\alpha<\lambda^+,\ \alpha\text{ a nonzero limit ordinal}\rangle \] such that each $\mathcal C_\alpha$ is a nonempty finite collection of closed unbounded subsets of $\alpha$, each member has order type at most $\lambda$, and whenever $C\in\mathcal C_\alpha$ and $\beta<\alpha$ is a limit point of $C$, one has $C\cap\beta\in\mathcal C_\beta$. There is no club $D\subseteq\lambda^+$ such that $D\cap\alpha\in\mathcal C_\alpha$ at every limit point $\alpha<\lambda^+$ of $D$. A limit point $\beta$ of a set $C$ satisfies $\beta>0$ and $\sup(C\cap\beta)=\beta$. Closedness means containing all limit points below the ambient ordinal, and unboundedness means being cofinal in that ordinal.

Extracted mathematical question. Given $\kappa$ with $V\models\mathrm{GCH}+\mathrm{SC}(\kappa)$, can one obtain a forcing extension $V[G]$ of the intended Prikry kind such that \[ V[G]\models\neg\square_{\aleph_\omega,<\omega}, \] without adding the hypothesis that $\kappa$ carries a normal measure? The cardinals in this displayed conclusion are evaluated in the extension. A normal measure on $\kappa$ means a nonprincipal $\kappa$-complete ultrafilter on $\kappa$ such that every regressive function on a measure-one set is constant on a measure-one subset. Omitting this hypothesis does not require assuming that no normal measure exists.

Unresolved scope. The original question is whether the assumption that $\kappa$ is measurable can be eliminated from Theorem 1. The supplied record does not reproduce that theorem or specify the intended forcing. Ordinary Prikry forcing uses a normal measure, so the phrase "Prikry extension" requires clarification when that hypothesis is removed. The record also does not specify the relationship between the ground-model $\kappa$ and the extension's $\aleph_\omega$, or any additional preservation requirements. Thus the displayed question records the supplied extracted target; a fully specified replacement for Theorem 1 cannot be recovered from this record alone.

Short English statement

Assuming GCH and the stated stationary-set principle $\mathrm{SC}(\kappa)$, can the intended Prikry extension make finite-width square fail at $\aleph_\omega$ without an additional measurability assumption on $\kappa$? The referenced theorem and the precise meaning of the extension are missing, so its full hypotheses and conclusion require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001684$, source problem code catalogue entry, title: Measurability Assumption in the Prikry Extension Theorem. The record assigns the problem to 2011 and supplies no individual proposer.
• Set Theory (2011), Oberwolfach Reports, pp. 85--140, DOI 10.4171/owr/2011/02: https://doi.org/10.4171/owr/2011/02. This is the supplied report citation and the literature-source link labelled "Oberwolfach problem statement." The original question refers to Theorem 1, whose text is absent from the supplied record.
• Supplied literature source labelled "Schimmerling and Zeman, square in core models": https://doi.org/10.2307/2687750.
• Supplied literature source labelled "consistency strength of failure of square": https://www.math.uci.edu/~mzeman/RESEARCH/ffs-0-120719.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the cited Prikry-extension theorem uses measurability of $\kappa$, with no elimination of that hypothesis found in the literature checked. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked to restore the large-cardinal principle, forcing conclusion, and measurability question. These are supplied verification claims; the linked sources have not been independently consulted for this draft.