Measurable-Cardinal Strength of Minimal Namba Collapse

Definitions and mathematical statement

Work over $\mathsf{ZFC}$. For a ground model $M$ and a set-forcing extension $M[G]$, ground-model ordinals are identified with the same ordinals in the extension. For regular cardinals $\nu,\mu$ of $M$, write \[ \mu=(\nu^{++})^M,\qquad \kappa=(\nu^+)^M,\qquad \lambda=(\mu^+)^M. \] The explicitly described collapse and preservation requirements, under a ground-model reading of the successor notation, are \[ M[G]\models |\mu|=\kappa \quad\text{and}\quad \operatorname{cf}^{M[G]}(\lambda)=\lambda. \] The second requirement says that the ground-model cardinal $\mu^+$ remains regular. The source additionally requires a "$|\mu|=\nu^+$-minimal" forcing extension. Its precise definition is not supplied, and this condition must not be omitted or replaced by an assumed definition of minimality. The source also does not explicitly settle whether all successor notation is evaluated in the ground model or which smaller cardinals must be preserved. Consequently, the displayed clauses describe the intended requirements only under the stated interpretation; they are not a complete definition of the source property.

Let $T_{\mathrm{src}}$ denote $\mathsf{ZFC}$ together with the assertion that there exist regular cardinals $\nu$ and $\mu=\nu^{++}$ and a set forcing yielding an extension with the source's exact minimality, collapse, and preservation properties. This notation is a placeholder for a theory whose full formulation requires the missing definition and hypotheses of the cited Theorem 1. The principal question is whether \[ \operatorname{Con}(T_{\mathrm{src}}) \ \Longrightarrow\ \operatorname{Con}\left(\mathsf{ZFC}+\text{there exists a measurable cardinal}\right). \] Here $\operatorname{Con}(T)$ means syntactic consistency of the theory $T$, and a measurable cardinal is an uncountable cardinal $\theta$ carrying a nonprincipal $\theta$-complete ultrafilter on $\theta$. This is a question about a consistency-strength lower bound. The source does not supply enough of Theorem 1 to state a matching upper-bound theorem precisely.

The original excerpt also asks how far the main theorem can be stratified, specifically whether a forcing like that of Theorem 1 can preserve cardinals $\nu\leq\lambda$. The parameter $\lambda$ in that excerpt is undefined and must not be identified with the auxiliary $\lambda$ above. This related question therefore remains incompletely specified.

The excerpt offers a speculative approach: if $\mu=\nu^{++}$ is not measurable in any inner model, it suggests obtaining a sequence of length $\mu^+$ witnessing "non-saturation of $\mu\cap\operatorname{cof}(\nu^+)$", and adapting an argument attributed to Shelah about singularizing $\mu$ while preserving $\mu^+$. The relevant ideal, hypotheses, and reference are absent; these remarks are motivation, not asserted theorems or additional assumptions. The excerpt expressly says that the role of minimality in such an argument is unclear.

Short English statement

Must the consistency of the specified minimal collapse of $\mu=\nu^{++}$ to size $\nu^+$, while preserving the regularity of the ground-model $\mu^+$, imply the consistency of a measurable cardinal? The source labels this question open. Its exact formulation requires the missing definition of minimality and the hypotheses of Theorem 1; the supplied excerpt also contains a separate, incompletely specified cardinal-preservation question.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006168$, exact source code catalogue entry, title "Measurable-Cardinal Strength of Minimal Namba Collapse." The supplied record gives proposed year 2025 and no named proposer.
• Supplied report citation: Set Theory (2025), Oberwolfach Reports, pp. 75--140, DOI 10.4171/owr/2025/2. https://doi.org/10.4171/owr/2025/2.
• Supplied literature source: Levine, On Namba Forcing and Minimal Collapses. https://resolve.cambridge.org/core/services/aop-cambridge-core/content/view/139989528588848BDDFD80F18AF63ED9/S2050509425101060a.pdf/on-namba-forcing-and-minimal-collapses.pdf.
• Supplied literature source: Oberwolfach Report 2/2025. https://content.ems.press/assets/public/full-texts/serials/owr/22/1/14299082/online/10.4171-owr-2025-2.pdf.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and says that Levine's paper retains the measurable-cardinal consistency question as an explicit remaining question, with the theorem providing only the current upper bound. This is supplied dated curation, not an independently verified literature assessment.
• The registry supplies statement status corrected_verified and says the statement was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft.
• The excerpt mentions Bukovský, Copláková, an argument attributed to Shelah, and references [4], [5, Sections 2.1 and 4.1], and [6]. Their bibliographic identities and precise supporting statements are not supplied.