Ben-Neria, Hayut, and Unger (J. Symbolic Logic 89, 2024) proved from a coherent Mitchell-increasing sequence of extenders with a supercompact cardinal between successive extender points that it is consistent for the Singular Cardinal Hypothesis (SCH) to fail at a singular strong limit cardinal of any cofinality while every collection of fewer than cf(ν)\mathrm{cf}(\nu) stationary subsets of ν+\nu^{+} reflects simultaneously, and asked whether the supercompactness could be removed (their Question 1.5). Two of the open clauses from that paper have since been settled: Benhamou and Sinapova (Trans. AMS, 2026) realized the pattern at ℵω1\aleph_{\omega_1} from uncountably many supercompact cardinals, answering the companion Question 1.3, and Oren (2026) obtained full, indeed countably simultaneous, stationary reflection at ℵω1+1\aleph_{\omega_1+1} from a single κ+\kappa^+-supercompact cardinal, though not together with the failure of SCH there. The surviving open core is the final clause of Question 1.5: is it consistent, from large-cardinal hypotheses at the level of a single partially supercompact cardinal and no supercompact cardinals, that κ\kappa becomes a strong limit singular cardinal of uncountable cofinality with 2κ>κ+2^\kappa > \kappa^+ while every stationary subset of κ+\kappa^+ reflects? Every known model of the combination at uncountable cofinality still uses supercompactness, and Oren (July 2026) explicitly lists the SCH-failure combination as work in progress.

Background

A subset SS of an ordinal δ\delta of uncountable cofinality is stationary if it meets every closed unbounded (club) subset of δ\delta; this is the standard notion of a large subset of δ\delta. SS reflects if some α<δ\alpha < \delta of uncountable cofinality has S∩αS \cap \alpha stationary in α\alpha, and a family of stationary sets reflects simultaneously if a single α\alpha works for all of them. Stationary reflection is a compactness principle: largeness is already witnessed below some smaller ordinal. The Singular Cardinal Hypothesis (SCH) asserts that for every strong limit singular cardinal ν\nu (singular: cf(ν)<ν\mathrm{cf}(\nu) < \nu; strong limit: 2μ<ν2^{\mu} < \nu for all μ<ν\mu < \nu) one has 2ν=ν+2^{\nu} = \nu^{+}. The failure of SCH at a singular cardinal is an anti-compactness phenomenon: it is known to require large cardinal axioms and to produce non-reflecting objects. Foreman and Todorcevic proved that if SCH fails at ν\nu then two stationary subsets of [ν+]ω[\nu^{+}]^{\omega} — the collection of countable subsets of ν+\nu^{+}, for which stationarity and reflection are defined analogously, via closed unbounded subfamilies — cannot reflect simultaneously; Shelah improved this to a single non-reflecting stationary subset of [ν+]ω[\nu^{+}]^{\omega}; and Poveda, Rinot, and Sinapova showed that the failure of SCH at ν\nu implies the failure of simultaneous reflection for collections of cf(ν)\mathrm{cf}(\nu) many stationary subsets of ν+\nu^{+}. Hence, while SCH fails at ν\nu, at best every collection of fewer than cf(ν)\mathrm{cf}(\nu) stationary subsets of ν+\nu^{+} can reflect simultaneously, and even single-set reflection at ν+\nu^{+} is delicate to combine with ¬SCH\neg\mathrm{SCH}.

The relevant large cardinal notions: κ\kappa is supercompact if for every λ\lambda there is an elementary embedding j:V→Mj : V \to M with critical point κ\kappa (the least ordinal moved by jj), j(κ)>λj(\kappa) > \lambda, and MM closed under λ\lambda-sequences; κ\kappa is κ+\kappa^{+}-supercompact if this holds for λ=κ+\lambda = \kappa^{+}. A (κ,λ)(\kappa, \lambda)-extender is the system of ultrafilters derived from such an elementary embedding with critical point κ\kappa and j(κ)≥λj(\kappa) \geq \lambda, coding the embedding up to λ\lambda; extenders are compared by the Mitchell order, E◃E′E \mathrel{\triangleleft} E' iff E∈Ult(V,E′)E \in \mathrm{Ult}(V, E'), the ultrapower of the universe by E′E'. The source paper does not formally define 'partially supercompact'; its own usage fixes the intended level. In Question 1.5 the phrase 'without any supercompactness assumptions' is equated, via 'Namely', with starting from 'a single partially supercompact cardinal κ\kappa', so what is excluded is full supercompactness — the supercompact cardinals between successive extender points in Theorem 1.1 — and any single-cardinal strength below it is admissible, paradigmatically κ+\kappa^{+}-supercompactness. The paper's abstract places the hypothesis of Theorem 1.4 even lower, reducing the consistency strength 'to below a single partially supercompact cardinal': that hypothesis is that κ\kappa is κ+\kappa^{+}-Π11\Pi^1_1-subcompact — every A⊆H(κ+)A \subseteq H(\kappa^{+}) (where H(θ)H(\theta) is the collection of sets whose transitive closure has size less than θ\theta) and every Π11\Pi^1_1 statement Φ\Phi (an assertion of the form 'for every XX, the first-order statement φ(X)\varphi(X) holds') true in ⟨H(κ+),∈,A⟩\langle H(\kappa^{+}), \in, A \rangle reflects, i.e. there are ρ<κ\rho < \kappa, B⊆H(ρ+)B \subseteq H(\rho^{+}) and an elementary embedding of ⟨H(ρ+),∈,B⟩\langle H(\rho^{+}), \in, B \rangle into ⟨H(κ+),∈,A⟩\langle H(\kappa^{+}), \in, A \rangle with critical point ρ\rho — possibly together with carrying a (κ,κ++)(\kappa, \kappa^{++})-extender; this is what the introduction calls 'subcompactness together with hypermeasurability', and it lies below κ+\kappa^{+}-supercompactness, itself far below supercompactness. All models below are built by Prikry-type forcing: forcing that adds a cofinal sequence of order type a prescribed regular cardinal η\eta to a large cardinal κ\kappa, making it singular of cofinality η\eta while controlling cardinal arithmetic and stationarity.

History of the combination. Magidor (1982) first showed, from ω\omega-many supercompact cardinals, that stationary reflection can hold at the successor of a singular cardinal (in a model of GCH, the generalized continuum hypothesis, i.e. 2μ=μ+2^{\mu} = \mu^{+} for every infinite cardinal μ\mu). Sharon (2005), from infinitely many supercompact cardinals, forced a singular ν\nu of countable cofinality at which SCH fails with every stationary subset of ν+\nu^{+} reflecting; Hayut and Unger (2020) later reduced the large-cardinal cost of reflection alone at successors of singulars to below a κ+\kappa^{+}-supercompact cardinal. Ben-Neria, Hayut, and Unger proved Theorem 1.1: from an increasing sequence ⟨κα∣α<η⟩\langle \kappa_\alpha \mid \alpha < \eta \rangle, each κα\kappa_\alpha carrying a (κα,λ)(\kappa_\alpha, \lambda)-extender, the sequence Mitchell-order increasing and coherent, and a supercompact cardinal between sup⁡β<ακβ\sup_{\beta<\alpha}\kappa_\beta and κα\kappa_\alpha for every α\alpha, there is a cardinal- and cofinality-preserving forcing extension in which, at κ=sup⁡ακα\kappa = \sup_\alpha \kappa_\alpha, one has 2κ=λ2^{\kappa} = \lambda and every collection of fewer than η\eta stationary subsets of κ+\kappa^{+} reflects simultaneously. For cf(κ)>ω\mathrm{cf}(\kappa) > \omega this was the first consistency result of its kind. Their Theorem 1.4 shows that at countable cofinality the supercompactness can be dropped: from a single κ+\kappa^{+}-Π11\Pi^1_1-subcompact cardinal carrying a (κ,κ++)(\kappa, \kappa^{++})-extender, one can force κ=ℵω\kappa = \aleph_\omega to be strong limit of cofinality ω\omega with 2κ=κ++2^{\kappa} = \kappa^{++} and every finite collection of stationary subsets of κ+\kappa^{+} reflecting simultaneously. Gitik gave an alternative proof of Theorem 1.1-type results with the same supercompact-based hypotheses via his overlapping-extender forcing, Benhamou and Sinapova brought the uncountable-cofinality pattern down to the small cardinal ℵω1\aleph_{\omega_1} from uncountably many supercompact cardinals, and Oren achieved full (indeed countably simultaneous) reflection at ℵω1+1\aleph_{\omega_1+1} from a single κ+\kappa^{+}-supercompact cardinal, though not together with the failure of SCH. What no construction has achieved is the combination — SCH failure together with stationary reflection at the successor — at uncountable cofinality from anything below supercompactness.

Problem Statement

Ben-Neria, Hayut, and Unger ask (Question 1.5 of their paper, quoted verbatim; the opening 'It is possible' is the paper's own typo for 'Is it possible'): 'It is possible to obtain the result of Theorem 1.1 without any supercompactness assumptions? Namely, is it possible to start with a single partially supercompact cardinal κ\kappa and force κ\kappa to be a strong limit singular of uncountable cofinality such that every stationary subset of κ+\kappa^{+} reflects? Is it possible to obtain that together with the failure of SCH at κ\kappa?' The reflection-only clause has since been answered positively by Oren (2026), who forces from a single κ+\kappa^{+}-supercompact cardinal that κ\kappa becomes ℵω1\aleph_{\omega_1} of cofinality ω1\omega_1 with countable simultaneous reflection at ℵω1+1\aleph_{\omega_1+1}. The surviving open core is the final clause:

Is it consistent, relative to large-cardinal hypotheses at the level of a single partially supercompact cardinal κ\kappa — for instance a κ+\kappa^{+}-supercompact cardinal, or a κ+\kappa^{+}-Π11\Pi^1_1-subcompact cardinal carrying a (κ,κ++)(\kappa, \kappa^{++})-extender — with no supercompact cardinals, that some forcing extension makes κ\kappa a strong limit singular cardinal of uncountable cofinality such that 2κ>κ+2^{\kappa} > \kappa^{+} (SCH fails at κ\kappa) and every stationary subset of κ+\kappa^{+} reflects? In the strong form matching the conclusion of Theorem 1.1: can every collection of fewer than cf(κ)\mathrm{cf}(\kappa) stationary subsets of κ+\kappa^{+} reflect simultaneously while SCH fails at κ\kappa, for example at κ=ℵω1\kappa = \aleph_{\omega_1}, mirroring at uncountable cofinality what Theorem 1.4 achieves at countable cofinality? Resolving either form constitutes a resolution: a positive answer to the single-set form with the failure of SCH is sufficient, while the fewer-than-cf(κ)\mathrm{cf}(\kappa)-fold simultaneous form is the desired optimum; either may be established at any uncountable cofinality, in particular at the instance κ=ℵω1\kappa = \aleph_{\omega_1}.

Since the configuration is already known to be consistent from supercompact cardinals, the problem is exactly one of consistency-strength calibration, and a negative answer at the partially supercompact level — an inner-model-theoretic argument showing that the consistency of the configuration implies the consistency of strictly stronger cardinals, such as a supercompact cardinal — would equally resolve it.

References

  1. Omer Ben-Neria, Yair Hayut, and Spencer Unger, Stationary reflection and the failure of the SCH, The Journal of Symbolic Logic 89 (2024), no. 1, 1-26, DOI 10.1017/jsl.2023.80, https://doi.org/10.1017/jsl.2023.80 (preprint: arXiv:1908.11145, https://arxiv.org/abs/1908.11145)
  2. Tom Benhamou and Dima Sinapova, Stationary reflection and the failure of SCH at ℵω1\aleph_{\omega_1}, Transactions of the American Mathematical Society (2026), DOI 10.1090/tran/9801, https://doi.org/10.1090/tran/9801 (preprint: arXiv:2411.09048, https://arxiv.org/abs/2411.09048)
  3. Inbar Oren, Stationary reflection using ancestrally forced conditions, preprint (2026), arXiv:2607.28350, https://arxiv.org/abs/2607.28350
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