HOD Failure at Uncountable Cofinality

Definitions and mathematical statement

Let $V$ denote the ambient universe of sets satisfying ZFC, and let $V_\alpha$ denote its cumulative hierarchy. All unqualified cardinals, cofinalities, power sets, and stationary sets are computed in $V$. Let $\mathrm{HOD}$ be the class of hereditarily ordinal-definable sets: those sets whose transitive closures consist of sets definable in $V$ from ordinal parameters.

For an uncountable regular cardinal $\delta$, put \[ S^\delta_\omega=\{\xi<\delta:\operatorname{cf}^{V}(\xi)=\omega\}. \] A subset of $\delta$ is stationary if it meets every closed unbounded subset of $\delta$ in $V$. Use the following standard formulation: $\delta$ is $\omega$-strongly measurable in $\mathrm{HOD}$ if there is an ordinal $\eta<\delta$ such that $(2^\eta)^{\mathrm{HOD}}<\delta$ and there is no sequence belonging to $\mathrm{HOD}$ that partitions $S^\delta_\omega$ into $\eta$ sets, each stationary in $V$. Here the power-set cardinality is computed inside $\mathrm{HOD}$. The HOD hypothesis says that there is a proper class of uncountable regular cardinals that are not $\omega$-strongly measurable in $\mathrm{HOD}$.

Use the following convention for $<\lambda$-extendibility: for every ordinal $\alpha$ with $\kappa<\alpha<\lambda$, there are an ordinal $\beta$ and an elementary embedding \[ j:(V_\alpha,\in)\longrightarrow(V_\beta,\in) \] with critical point $\kappa$ and $j(\kappa)>\alpha$. No requirement $\beta<\lambda$ is imposed in this convention. The supplied record does not specify the definition of this term, so agreement with the source's intended convention requires review.

A cardinal $\lambda$ is a strong limit if $2^\mu<\lambda$ for every cardinal $\mu<\lambda$. Write $\lambda^+$ for the least cardinal of $V$ strictly above $\lambda$, and write $(\lambda^+)^{\mathrm{HOD}}$ for the least cardinal strictly above the ordinal $\lambda$ as computed in $\mathrm{HOD}$.

Consistency question. Is the following theory consistent? \[ \mathrm{ZFC}+\text{the HOD hypothesis} +\exists\text{ cardinals }\kappa<\lambda\; \left[ \begin{gathered} \kappa\text{ is }<\lambda\text{-extendible},\\ \lambda\text{ is a strong-limit cardinal},\\ \operatorname{cf}^{V}(\lambda)=\omega_1^{V},\\ (\lambda^+)^{\mathrm{HOD}}<\lambda^+ \end{gathered} \right]. \] No relative consistency assumption is specified in the supplied record. This formulation follows its reconstructed statement. The original excerpt asks whether "the above configuration" is consistent, but does not include that preceding configuration; consequently, the reconstruction cannot be checked against the excerpt alone.

Short English statement

Can a universe satisfying ZFC and the HOD hypothesis contain cardinals $\kappa<\lambda$ such that $\kappa$ is extendible up to every rank below $\lambda$, $\lambda$ is a strong limit of cofinality $\omega_1$, and HOD computes a strictly smaller successor cardinal of $\lambda$ than the ambient universe does? The supplied reconstruction is used, with its extendibility convention requiring source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006167$, title HOD Failure at Uncountable Cofinality, exact source code catalogue entry. The original excerpt labels the question "Question 10 (Goldberg-P.)." The supplied proposer field is empty.
• Set Theory (2025), Oberwolfach Reports, pp. 75--140, DOI 10.4171/owr/2025/2. Supplied report reference: https://doi.org/10.4171/owr/2025/2.
• Goldberg--Poveda, Compactness phenomena in HOD. Supplied literature reference: https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/compactness-phenomena-in-hod/FFD6AD92EFDDDAC2D53A98CDE7A396AD.
• Oberwolfach Report 2/2025. Supplied full-text reference: https://content.ems.press/assets/public/full-texts/serials/owr/22/1/14299082/online/10.4171-owr-2025-2.pdf.
• Supplied literature assessment dated 2026-08-22: the 2025 primary paper reportedly still presents the analogous HOD failure configuration at uncountable cofinality, including cofinality $\omega_1$, as an open consistency question. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The registry labels the problem open and the statement corrected_verified, and reports that the reconstruction was checked against the official report; these are supplied metadata claims, not verification performed for this draft.