Consistency strength of the higher Solovay configuration at singular cardinals of countable cofinality: measurability of $\kappa^+$ in $L(\mathcal{P}(\kappa))$ and the $\aleph_\omega$-perfect set property

The inner model $L(\mathcal{P}(\kappa))$, the smallest transitive model of ZF containing the full powerset of a cardinal $\kappa$, is where $2^\kappa$ lives; at singular cardinals of countable cofinality it can satisfy choiceless theories strong enough to destroy the pcf-theoretic structure (scales, SCH, approachability) that ZFC provides. Poveda and Thei constructed, from a supercompact-type hypothesis via Merimovich's Prikry forcing, a higher Solovay model in which $L(\mathcal{P}(\aleph_\omega))$ satisfies ZF, dependent choices of length $\aleph_\omega$, and the $\aleph_\omega$-perfect set property for every subset of the generalized Baire space ${}^{\omega}\aleph_\omega$, together with the failure of scales, SCH, diamond and approachability. That work leaves three linked calibration questions open: the exact consistency strength of the assertion that $L(\mathcal{P}(\kappa))$ satisfies '$\kappa^+$ is measurable' for singular $\kappa$ of countable cofinality (upper bound: the rank-into-rank axiom $I_0(\kappa)$, by a theorem of Woodin); whether $\aleph_{\omega+1}$ can be measurable in $L(\mathcal{P}(\aleph_\omega))$ at all, a configuration currently not known to be consistent since no $\aleph_\omega$-analogue of $I_0$ exists; and the exact consistency strength of ZF + DC$_{\aleph_\omega}$ + the $\aleph_\omega$-perfect set property for all subsets of ${}^{\omega}\aleph_\omega$, which is bounded above by a supercompact cardinal with an inaccessible above and, from below, on its definable fragments only, by inner-model hypotheses around $0^\dagger$.

Background

A cardinal $\kappa$ is singular when it is the supremum of fewer than $\kappa$ smaller cardinals; its cofinality $\mathrm{cf}(\kappa)$ is the least order type of an unbounded sequence below it. The first singular cardinal is $\aleph_\omega=\sup_{n<\omega}\aleph_n$, and it has countable cofinality. In ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) the combinatorics of such cardinals is governed by Shelah's pcf theory ('possible cofinalities'): ZFC proves that every singular cardinal carries a scale, that is, a cofinal increasing sequence of length $\kappa^+$ in a product of smaller cardinals modulo eventual domination; the Singular Cardinals Hypothesis (SCH), asserting that $2^\kappa=\kappa^+$ for every strong limit singular $\kappa$, can fail only from very large cardinals; and Shelah's approachability property $\mathrm{AP}_\kappa$ organizes the structure of $\kappa^+$ through a coherent sequence of club sets. All of these objects are artifacts of the Axiom of Choice, and the question of what replaces them in models without Choice is a distinct, and largely open, subject.

The template comes from classical descriptive set theory. A set of reals has the perfect set property (PSP) if it is countable or contains a nonempty perfect subset, and a classical construction under the Axiom of Choice produces sets of reals without it. Under the Axiom of Determinacy (AD) every set of reals has the PSP, and AD holds in $L(\mathbb{R})$, the smallest transitive model of ZF containing the reals, only from very large cardinals; Solovay's 1970 model, by contrast, obtains regularity for all sets of an inner model from a single inaccessible cardinal. Calibrating the exact large-cardinal strength of such regularity theories is one of the central themes of modern set theory.

Generalized descriptive set theory (GDST) transports this study beyond the continuum. For a cardinal $\kappa$ of countable cofinality, the role of the reals is played by the generalized Baire space ${}^{\omega}\kappa$ of countable sequences of ordinals below $\kappa$, and a set $A\subseteq{}^{\omega}\kappa$ has the $\kappa$-perfect set property ($\kappa$-PSP) if either $|A|\le\kappa$ or $A$ contains a topologically embedded copy of the generalized Cantor space ${}^{\kappa}2$, closed in ${}^{\omega}\kappa$. The natural choiceless inner model is $L(\mathcal{P}(\kappa))$, the smallest transitive model of ZF containing the full powerset of $\kappa$. When the cofinality of $\kappa$ is uncountable, a theorem of Shelah (invoked in Poveda-Thei) gives the Axiom of Choice inside $L(\mathcal{P}(\kappa))$, so the $\kappa$-PSP fails there; countable cofinality is therefore the interesting case. The strongest large-cardinal axiom not ruled out by the classical theorem forbidding nontrivial elementary embeddings of the universe into itself is the rank-into-rank axiom $I_0(\kappa)$: there is an elementary embedding $j:L(V_{\kappa+1})\to L(V_{\kappa+1})$ with critical point (the least ordinal moved by $j$) below $\kappa$. A theorem of Woodin, recorded as Theorem 2.9 in the work of Poveda-Thei, shows that under $I_0(\kappa)$ the model $L(V_{\kappa+1})$ satisfies both the $\kappa$-PSP for every subset of ${}^{\omega}\kappa$ and the measurability of $\kappa^+$, and Shi-Trang derived from $I_0(\kappa)$ that $L(\mathcal{P}(\kappa))$ has no scales at $\kappa$ and no $\kappa^+$-sequence of distinct subsets of $\kappa$. Thus $I_0(\kappa)$ plays for $L(V_{\kappa+1})$ the role that determinacy plays for $L(\mathbb{R})$.

The $I_0$ picture, however, lives at enormously large cardinals: $I_0(\kappa)$ requires $\kappa$ to be a limit of very large cardinals and so can never hold at the first singular cardinal $\aleph_\omega$. Poveda and Thei, building on their precursor work with Dimonte, asked whether the same configuration can nonetheless be reached at $\aleph_\omega$ by forcing. Replacing the Levy collapse of Solovay's classical construction with Merimovich's supercompact extender based Prikry forcing, a forcing that changes the cofinality of a $<\lambda$-supercompact cardinal $\kappa$ to $\omega$ while making an inaccessible $\lambda$ become $\kappa^+$, they construct a generic extension in which $L(\mathcal{P}(\kappa))$ satisfies ZF, dependent choices of length $\kappa$ ($\mathrm{DC}_\kappa$), the $\kappa$-PSP for every subset of ${}^{\omega}\kappa$, the failure of SCH and of diamond at $\kappa^+$, the non-existence of scales at $\kappa$, the failure of the approachability property, and additional partition and tree properties. This answered, in the choiceless context of ZF + $\mathrm{DC}_{\aleph_\omega}$, a question dating from the 1980s on whether $\aleph_\omega$ can be strong limit while SCH and approachability both fail, and a 2024 question of Woodin on whether the Cramer-Shi-Woodin regularity configuration is consistent at the first singular cardinal at all.

What the construction does not produce is measurability of $\kappa^+$ inside $L(\mathcal{P}(\kappa))$ itself: adjoining a ground-model normal measure $U$ gives a model $L(\mathcal{P}(\kappa),U)$ in which $\kappa^+$ is measurable, but without $U$ no such conclusion is available. The three questions considered here are the calibration problems this leaves open: how much large-cardinal strength the choiceless structure of $L(\mathcal{P}(\kappa))$ at a singular of countable cofinality really requires, and whether the strongest part of the $I_0$ picture, a measurable successor, is consistent at all at the first singular cardinal.

Problem Statement

Three linked questions, posed as Questions 4.1-4.3 of Poveda-Thei (arXiv:2509.18991v2). Throughout, $\aleph_\omega=\sup_{n<\omega}\aleph_n$ is the first singular cardinal; $L(\mathcal{P}(\kappa))$ is the smallest transitive model of ZF containing the full powerset of $\kappa$; ${}^{\omega}\kappa$ is the generalized Baire space, the set of countable sequences of ordinals below $\kappa$; a set $A\subseteq{}^{\omega}\kappa$ has the $\kappa$-perfect set property ($\kappa$-PSP) if either $|A|\le\kappa$ or $A$ contains a copy of the generalized Cantor space ${}^{\kappa}2$ that is closed in ${}^{\omega}\kappa$; and when a cardinal is said to be measurable in an inner model, the cardinal itself (for instance $\kappa^+$ or $\aleph_{\omega+1}$) is computed in the ambient universe, while the witnessing $\mu$-complete nonprincipal ultrafilter on it is required to be an element of the inner model.

(1) [Question 4.1, attributed by the source to Dimonte-Poveda-Thei] Assume $\kappa$ is a singular cardinal of countable cofinality. What is the exact consistency strength of the assertion that $L(\mathcal{P}(\kappa))\models$ '$\kappa^+$ is measurable', that is, with which standard large-cardinal hypothesis is the consistency of this configuration equiconsistent? An upper bound is the rank-into-rank axiom $I_0(\kappa)$, which asserts the existence of an elementary embedding $j:L(V_{\kappa+1})\to L(V_{\kappa+1})$ with critical point below $\kappa$, by a theorem of Woodin; the matching lower bound, and hence the exact calibration, is unknown.

(2) [Question 4.2] Is it consistent that $\aleph_{\omega+1}$ is measurable in $L(\mathcal{P}(\aleph_\omega))$? This is the instance of (1) at the first singular cardinal, where the configuration is not even known to be consistent: $I_0$-type axioms require their cardinal to be a limit of very large cardinals and thus cannot hold at $\aleph_\omega$, and no alternative upper-bound axiom or construction is known.

(3) [Question 4.3] What is the exact consistency strength of the theory

$$\mathrm{ZF}+\mathrm{DC}_{\aleph_\omega}+\text{every }A\subseteq{}^{\omega}\aleph_\omega\text{ has the }\aleph_\omega\text{-perfect set property},$$

where $\mathrm{DC}_{\aleph_\omega}$ is the axiom of dependent choices of length $\aleph_\omega$ (any binary relation that admits a next element after every sequence of fewer than $\aleph_\omega$ successive choices admits a full $\aleph_\omega$-sequence of successive choices)? The current upper bound is a $<\lambda$-supercompact cardinal with an inaccessible $\lambda$ above, via Merimovich's forcing; the known lower bounds concern only definable fragments of the pointclass (currently the existence of $0^\dagger$, the sharp for an inner model with a measurable cardinal of positive Mitchell order, from the boldface $\kappa$-$\Pi^1_1$ fragment).

The common objective is to calibrate, in the standard equiconsistency sense, how much large-cardinal structure the powerset of a singular cardinal of countable cofinality can carry, the choiceless counterpart of the calibration of determinacy in $L(\mathbb{R})$ by infinitely many Woodin cardinals. The parts are linked: (2) is the strongest specialization of (1), at the cardinal where even consistency is open, and (3) calibrates the regularity property which the source line of work shows already implies most of the combinatorial consequences of $I_0(\kappa)$ inside $L(\mathcal{P}(\kappa))$.

References

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