Optimal large-cardinal strength for blowing up the power of a singular cardinal of uncountable cofinality that is singular in the core model

Gitik (J. Symbolic Logic, 2019) constructed the first cardinal-preserving forcing that blows up the power of a singular cardinal of uncountable cofinality which is singular in the core model, starting from a ground model carrying a coherent, Mitchell-increasing (overlapping) sequence of extenders $\langle E(\alpha)\mid\alpha<\eta\rangle$ with $E(\beta)\triangleleft E(\alpha)$ — a hypothesis of roughly the strength of $\eta$-many strong cardinals, which Gitik conjectured to be optimal — together with a reflection condition $j_{E(\alpha)}(h^\alpha_\lambda)(\kappa_\alpha)=\lambda$ assumed for simplicity in the construction. It remains open to determine the exact consistency strength of such blow-ups: no lower bound deriving a comparable extender sequence from a model with the blow-up configuration is known, since the Gitik–Mitchell covering-lemma analysis of the core model breaks down precisely in the presence of overlapping extenders.

Background

The Singular Cardinal Hypothesis (SCH) asserts that if $\kappa$ is a singular strong limit cardinal (an uncountable limit cardinal with $2^\alpha<\kappa$ for all $\alpha<\kappa$), then $2^\kappa=\kappa^+$. While Easton's theorem leaves the power-set function free on regular cardinals subject to monotonicity and König's theorem, the behaviour at singular cardinals is tightly constrained: Silver's theorem implies that SCH cannot fail for the first time at a singular cardinal of uncountable cofinality, so a forcing that blows up $2^\kappa$ at a singular $\kappa$ of uncountable cofinality $\eta$ must simultaneously arrange blown-up powers cofinally below $\kappa$. This is why such blow-ups require large cardinals, and why the countable- and uncountable-cofinality cases have very different characters.

The classical calibration machinery combines forcing with inner model theory. An extender is a system of ultrafilters coding an elementary embedding $j:V\to M$ with critical point $\operatorname{crit}(j)=\kappa$; a $(\kappa,\lambda)$-extender witnesses that $\kappa$ is $\lambda$-strong (that is, $j(\kappa)>\lambda$, with $M$ closed under $\kappa$-sequences in the versions used here). The core model $K$ is the canonical inner model below the large-cardinal hypothesis in question, and the covering lemma for $K$ states roughly that every uncountable set of ordinals is covered by a set of the same cardinality lying in $K$, possibly enlarged by a sequence of indiscernibles; from it one reads off lower bounds for failures of SCH in terms of the Mitchell order $o^K(\kappa)$, which measures how many measures or extenders $K$ carries on $\kappa$. Gitik and Mitchell (1996) proved that if $\kappa$ is singular strong limit with $\operatorname{cf}(\kappa)=\delta$ and $2^\kappa\ge\lambda>\kappa^+$, with $\lambda$ not the successor of a cardinal of cofinality below $\kappa$, then within the core-model framework for sequences of non-overlapping extenders, $\delta>\omega_1$ implies $o^K(\kappa)\ge\lambda+\delta$ and $\delta=\omega_1$ implies $o^K(\kappa)\ge\lambda$; a construction of Woodin reported in that paper yields $2^\kappa=\lambda$ with $\operatorname{cf}(\kappa)=\delta>\omega$ from $o(\kappa)=\lambda+\delta$. In all of these models, however, $\kappa$ is measurable in $K$ (a measurable cardinal is one carrying a nontrivial $\kappa$-complete ultrafilter, i.e. the critical point of an elementary embedding) and is made singular by Prikry-type forcing, which adds a cofinal sequence through a measurable cardinal. The regime in which $\kappa$ is itself singular in the core model was out of reach: the indiscernible-sequence analysis behind every known lower bound breaks down exactly when $K$ contains overlapping extenders, that is, extenders one of which is an initial segment of the other.

Gitik's 2019 paper supplies the missing upper bound. Its Theorem 1.1 assumes GCH, a regular cardinal $\eta$, and an increasing sequence $\langle\kappa_\alpha\mid\alpha<\eta\rangle$ with $\kappa_0>\eta$ carrying extenders $E(\alpha)$ with critical points $\kappa_\alpha$ and $j_{E(\alpha)}(\kappa_\alpha)>\lambda$ for a prescribed regular $\lambda>\bigcup_{\alpha<\eta}\kappa_\alpha$, arranged so that $E(\beta)\triangleleft E(\alpha)$ whenever $\beta<\alpha$ (each earlier extender is an initial segment of each later one; in particular the sequence is Mitchell-increasing, i.e. $E_\beta$ lies in the ultrapower by $E_\alpha$ for $\beta<\alpha$, in the strongest coherent way). This is a hypothesis of the strength of $\eta$-many strong cardinals (a cardinal is strong if it is $\theta$-strong for every $\theta$), or slightly less, and it is this hypothesis that Gitik conjectures to be optimal. For the construction the paper additionally assumes, for simplicity, a reflection requirement $j_{E(\alpha)}(h^\alpha_\lambda)(\kappa_\alpha)=\lambda$ reflecting the target $\lambda$ below $\kappa_\alpha$; under a Woodin cardinal the coherent sequence can be picked so as to satisfy it. The associated forcing combines the one-element extender-based Prikry forcings of the individual extenders into a single poset with a direct order $\le^*$ and a forcing order $\le$; the paper proves the Prikry property (every statement about the generic object is decided by a direct extension that adds no new points), a $\bar\kappa_\eta^{++}$-chain condition (no antichain of that size, which prevents collapsing cardinals), and closure of the tail forcings. Consequently all cardinals and cofinalities are preserved while $\bar\kappa_\eta=\bigcup_{\alpha<\eta}\kappa_\alpha$ becomes a strong limit singular cardinal of cofinality $\eta$ with $2^{\bar\kappa_\eta}=\lambda$; this is the first cardinal-preserving blow-up of a singular of uncountable cofinality that is singular in the core model. A variant of the construction with $\eta$ inaccessible blows up the powers of a club class of singular cardinals below $\eta$.

Later work built on the method without settling its cost. Ben-Neria, Hayut and Unger (2024) obtained, from a Mitchell-order-increasing and coherent extender sequence supplemented by a supercompact cardinal between successive critical points (a supercompact cardinal is one whose embedding closes the target model under arbitrarily long sequences), a cardinal-preserving extension with $2^\kappa=\lambda$ together with simultaneous reflection of stationary subsets of $\kappa^+$ (a stationary set meets every closed unbounded set; reflection means it is already stationary below some ordinal) — a hypothesis stronger than Gitik's. Gitik himself gave direct models of $\neg$SCH with reflection (2022) and a new method for $\neg$SCH with the failure of approachability, a covering-type combinatorial principle at successors of singulars (2021). Jirattikansakul (2023) weakened the large-cardinal assumption to a Mitchell-increasing sequence of extenders of bounded length $\lambda=\rho^{+n}$, where $\rho=\sup_{\alpha<\eta}\kappa_\alpha$, an assumption he characterizes as below a Woodin cardinal (a Woodin cardinal is a cardinal that is a limit of cardinals that are $\theta$-strong for every $\theta$ below it), but his forcing interleaves collapses: it preserves cardinals and cofinalities only up to and including $\eta$, the ground-model singular moves up to $\aleph_{\gamma+\eta}$ (where $\eta^+=\aleph_\gamma$), and the cardinals in between are collapsed. What no later work provides is a matching lower bound, or a cardinal-preserving construction from a genuinely weaker hypothesis: the exact strength of the configuration, conjectured by Gitik to be precisely his overlapping-extender hypothesis, remains unknown.

Problem Statement

Within the framework of Gitik's overlapping-extender forcing (J. Symbolic Logic 84 (2019) 1722-1743), the following is open.

Optimality, i.e. exact consistency strength. Let $C$ be the assertion that there is a cardinal-preserving forcing extension in which some singular strong limit cardinal $\kappa$ of uncountable cofinality $\eta$, which is singular in the core model (that is, $\kappa$ is itself singular in the core model $K$, in contrast to the previously analyzed case of a cardinal measurable in $K$ that is singularized only by Prikry-type forcing), has $2^\kappa>\kappa^+$ (in the source theorem, $2^\kappa=\lambda$ for an arbitrarily prescribed regular $\lambda>\kappa$). Gitik derives $C$ from a ground model satisfying GCH and carrying a regular cardinal $\eta$, an increasing sequence $\langle\kappa_\alpha\mid\alpha<\eta\rangle$ with $\kappa_0>\eta$, and a sequence of extenders $\langle E(\alpha)\mid\alpha<\eta\rangle$ such that each $E(\alpha)$ is a $(\kappa_\alpha,\lambda)$-extender (with $\operatorname{crit}(j_{E(\alpha)})=\kappa_\alpha$, $j_{E(\alpha)}(\kappa_\alpha)>\lambda$, $M_{E(\alpha)}\supseteq H_\lambda$, and ${}^{\kappa_\alpha}M_{E(\alpha)}\subseteq M_{E(\alpha)}$) and $E(\beta)\triangleleft E(\alpha)$ for every $\beta<\alpha$ — a Mitchell-increasing, overlapping sequence of roughly the strength of $\eta$-many strong cardinals — and conjectures this hypothesis to be optimal for obtaining failures of SCH at a singular cardinal of uncountable cofinality that is singular in the core model. (In the forcing construction, a reflection property $j_{E(\alpha)}(h^\alpha_\lambda)(\kappa_\alpha)=\lambda$ is additionally assumed for simplicity; under a Woodin cardinal the coherent extender sequence can be picked so as to satisfy it.) Prove or refute this conjecture: either show that every model of $C$ has an inner model carrying such an overlapping-extender sequence (or an inner model of a hypothesis of the same consistency strength), thereby fixing the exact consistency strength of $C$; or exhibit $C$ via a cardinal-preserving forcing from a strictly weaker large-cardinal hypothesis.

References

1. Moti Gitik, Blowing up the power of a singular cardinal of uncountable cofinality, The Journal of Symbolic Logic 84(4) (2019) 1722-1743, DOI 10.1017/jsl.2019.66, https://doi.org/10.1017/jsl.2019.66
2. Moti Gitik and William J. Mitchell, Indiscernible sequences for extenders, and the singular cardinal hypothesis, Annals of Pure and Applied Logic 82(3) (1996) 273-316, DOI 10.1016/s0168-0072(96)00007-3, arXiv:math/9507214, https://doi.org/10.1016/s0168-0072(96)00007-3
3. Sittinon Jirattikansakul, Blowing up the power of a singular cardinal of uncountable cofinality with collapses, Annals of Pure and Applied Logic 174(6) (2023) 103257, DOI 10.1016/j.apal.2023.103257, arXiv:2011.00409, https://doi.org/10.1016/j.apal.2023.103257
4. Omer Ben-Neria, Yair Hayut, and Spencer Unger, Stationary reflection and the failure of the SCH, The Journal of Symbolic Logic 89(1) (2024) 1-26, DOI 10.1017/jsl.2023.80, arXiv:1908.11145, https://doi.org/10.1017/jsl.2023.80
5. Moti Gitik, Reflection and not SCH with overlapping extenders, Archive for Mathematical Logic 61(5-6) (2022) 591-597, DOI 10.1007/s00153-021-00805-3, https://doi.org/10.1007/s00153-021-00805-3
6. Moti Gitik, Another method for constructing models of not approachability and not SCH, Archive for Mathematical Logic 60(3-4) (2021) 469-475, DOI 10.1007/s00153-020-00755-2, https://doi.org/10.1007/s00153-020-00755-2
7. Alejandro Poveda, Assaf Rinot, and Dima Sinapova, Sigma-Prikry forcing III: Down to $\aleph_\omega$, Advances in Mathematics 435 (2023) 109377, DOI 10.1016/j.aim.2023.109377, arXiv:2209.10501, https://doi.org/10.1016/j.aim.2023.109377
8. Moti Gitik and Menachem Magidor, Extender based forcings, The Journal of Symbolic Logic 59(2) (1994) 445-460, DOI 10.2307/2275399, https://doi.org/10.2307/2275399
9. Moti Gitik and Sittinon Jirattikansakul, Extender-based Magidor-Radin forcings without top extenders, Annals of Pure and Applied Logic 176(8) (2025) 103584, DOI 10.1016/j.apal.2025.103584, arXiv:2306.12831, https://doi.org/10.1016/j.apal.2025.103584