The Revision Hypothesis: must every limit cardinal be a revision cardinal?

Shelah's Revised GCH theorem, a cornerstone of pcf theory, proves in ZFC that every uncountable strong limit cardinal is a "revision cardinal": for every $\lambda\ge\chi$ the covering number $\operatorname{cov}(\lambda,\chi,\chi,\sigma)$ is at most $\lambda$ for some $\sigma<\chi$. Matet's Revision Hypothesis asks whether the strong-limit assumption can be dropped: is every limit cardinal a revision cardinal? By a reduction recorded in the source paper this is equivalent to asking whether, for any two singular cardinals $\chi<\lambda$ of countable cofinality, $\operatorname{cov}(\lambda,\chi,\chi,\sigma)\le\lambda$ holds for some $\sigma<\chi$. The hypothesis follows from Shelah's Medium Hypothesis, holds under Shelah's Strong Hypothesis (hence under GCH) and in all known proximity regimes, and its failure would require large cardinals; yet neither a ZFC proof nor a consistency proof of failure is known.

Background

In ZFC the continuum function $\kappa\mapsto 2^\kappa$ is essentially unconstrained on regular cardinals (Easton's theorem), while at singular cardinals the behavior of exponentiation is governed by Shelah's pcf theory. For a set $A$ of regular cardinals, $\operatorname{pcf}(A)$ is the set of cardinals arising as the true cofinality $\operatorname{tcf}(\prod A/I)$ of the reduced product of $A$ modulo a proper ideal $I$ on $A$, i.e. the least length of an increasing cofinal sequence in that reduced product modulo $I$ when one exists. Controlling the size and structure of $\operatorname{pcf}(A)$ is the engine behind ZFC bounds on singular cardinal arithmetic, such as Shelah's bound $2^{\aleph_\omega}<\aleph_{\omega_4}$ when $\aleph_\omega$ is a strong limit.

A convenient packaging of pcf conclusions uses covering numbers. For cardinals $\rho_1,\rho_2,\rho_3,\rho_4$ with $\rho_1^+\ge\rho_2$, $\rho_2^+\ge\rho_3\ge\omega$ and $\rho_3\ge\rho_4\ge 2$, $\operatorname{cov}(\rho_1,\rho_2,\rho_3,\rho_4)$ is the least size of a family $X\subseteq\mathcal P_{\rho_2}(\rho_1)$ — where $\mathcal P_\tau(S)=\{x\subseteq S:|x|<\tau\}$ — such that every $b\in\mathcal P_{\rho_3}(\rho_1)$ is a union of fewer than $\rho_4$ members of $X$. Following Matet, a cardinal $\chi\le\lambda$ is a $\lambda$-revision cardinal if $\operatorname{cov}(\lambda,\chi,\chi,\sigma)\le\lambda$ for some $\sigma<\chi$, and $\chi$ is a revision cardinal if it is a $\lambda$-revision cardinal for every $\lambda\ge\chi$.

Shelah's Revised GCH theorem (RGCH), proved in ZFC in "The generalized continuum hypothesis revisited" (Israel Journal of Mathematics 116 (2000)), states that every uncountable strong limit cardinal $\chi$ (i.e. $2^{\theta}<\chi$ for every cardinal $\theta<\chi$) is a revision cardinal. In the equivalent revised-power-set formulation used in Eisworth's exposition: if $\mu$ is an uncountable strong limit cardinal then for every $\lambda\ge\mu$ there is $\kappa_0<\mu$ such that $\lambda^{[\kappa]}=\lambda$ for all $\kappa$ with $\kappa_0\le\kappa<\mu$, where $\lambda^{[\kappa]}$ is the least size of a family $P\subseteq[\lambda]^{\le\kappa}$ such that every $\kappa$-subset of $\lambda$ is a union of fewer than $\kappa$ members of $P$. This is one of the deepest ZFC theorems about singular cardinals and underlies Shelah's applications to diamond principles and covering numbers; Eisworth later gave a proof that eliminates the "nice filter" machinery — and with it any appeal to large cardinals in core models (the canonical inner models for large cardinals) — used in one of Shelah's arguments.

The strong limit hypothesis is a serious restriction, and it is exactly what supplies the counting input $|\operatorname{pcf}(A)|\le 2^{|A|}<\min(A)$ for the sets $A$ of regular cardinals in the interval $(\chi,\lambda)$; Shelah's theorem was in fact proved from the weaker pcf hypothesis $|\operatorname{pcf}(A)|<\mu$. Shelah long sought conditions under which $\beth_\omega$ could be replaced by $\aleph_\omega$ in such statements (as recounted in the source paper), and Matet's paper pushes the question to its natural limit under the name Revision Hypothesis: every limit cardinal — not just every strong limit cardinal — is a revision cardinal. By a stationarity argument (Fact 2.1(ii) of the source), the hypothesis holds iff $\chi$ is a $\lambda$-revision cardinal whenever $\chi<\lambda$ are singular cardinals of countable cofinality; this is the essential open case.

The hypothesis sits in a web of better-known statements. Shelah's Medium Hypothesis (SMH) asserts $|\operatorname{pcf}(A)|\le|A|$ whenever $\min(A)>|A|$; Shelah's Weak Hypothesis (SWH) asserts that below any cardinal $\lambda$ there are only countably many singular $\theta$ with $\operatorname{pp}(\theta)\ge\lambda$, where $\operatorname{pp}(\theta)$, the pseudopower of a singular $\theta$, is the supremum of the true cofinalities of reduced products of sets of regular cardinals cofinal in $\theta$. Gitik and Shelah proved SWH implies SMH, and SMH — indeed the weaker bound $|\operatorname{pcf}(A)|<|A|^{+\omega}$ — implies the Revision Hypothesis. On the other side, the hypothesis holds under Shelah's Strong Hypothesis SSH (the assertion $\operatorname{pp}(\theta)=\theta^+$ for every singular $\theta$), hence under GCH: under SSH, for $\chi\le\lambda$, the only cardinal that fails to be a $\lambda$-revision cardinal is the successor cardinal $(\operatorname{cf}\lambda)^+$, which is never a limit cardinal. But SSH and SWH do fail in forcing extensions built from large cardinals (Gitik), and SMH fails in Gitik's short-extender models (forcing constructions built from extenders, the elementary embeddings supplied by large cardinals), where a countable $A$ can have $\operatorname{pcf}(A)$ of order type $\omega_1+1$ — although even there $|\operatorname{pcf}(A)|<|A|^{+\omega}$ survives, so the Revision Hypothesis is undecided in every known model. In ZFC the hypothesis is confirmed only in proximity regimes, for instance $\chi$ is a $\lambda$-revision cardinal whenever $\chi<\lambda<\chi^{+\chi}$, and in various regimes where $\chi$ is not a fixed point of the aleph function (that is, $\chi\neq\aleph_\chi$).

Problem Statement

The Revision Hypothesis (Matet, "Meeting, covering and Shelah's Revised GCH", arXiv:2308.14779, Section 2.2): every limit cardinal is a revision cardinal. Explicitly, decide in ZFC whether the following holds:

$$\text{For every limit cardinal }\chi\text{ and every cardinal }\lambda\ge\chi\text{ there exists a cardinal }\sigma<\chi\text{ such that }\operatorname{cov}(\lambda,\chi,\chi,\sigma)\le\lambda.$$

By Fact 2.1(ii) of the source paper, together with the fact that the singular cardinals below any limit cardinal of uncountable cofinality form a stationary set (one meeting every closed unbounded subset), this is equivalent to the same statement restricted to pairs $\chi<\lambda$ of singular cardinals of countable cofinality, which is the essential case. For strong limit $\chi$ the conclusion is Shelah's Revised GCH theorem, so the problem concerns precisely the limit cardinals that are not strong limits.

A ZFC proof that every limit cardinal is a revision cardinal would resolve the question affirmatively; a model of ZFC containing a limit cardinal $\chi$ and a cardinal $\lambda\ge\chi$ such that $\operatorname{cov}(\lambda,\chi,\chi,\sigma)>\lambda$ for every $\sigma<\chi$, relative to stated large-cardinal hypotheses, would resolve it negatively.

References

1. Pierre Matet, "Meeting, covering and Shelah's Revised GCH", arXiv:2308.14779 (2023); published as Topology and its Applications 356 (2024), article 109044, DOI 10.1016/j.topol.2024.109044, https://arxiv.org/abs/2308.14779
2. Pierre Matet, "A short tour of Shelah's Revised GCH Theorem", arXiv:2308.14467 (2023), https://arxiv.org/abs/2308.14467
3. Pierre Matet, "The pcf theory of non fixed points", arXiv:2308.15511 (2023), https://arxiv.org/abs/2308.15511
4. Saharon Shelah, "The generalized continuum hypothesis revisited", Israel Journal of Mathematics 116 (2000), 285-321, DOI 10.1007/BF02773223, https://doi.org/10.1007/bf02773223
5. Saharon Shelah, "On what I do not understand (and have something to say): Part I", Fundamenta Mathematicae 166 (2000), 1-82, DOI 10.4064/fm-166-1-2-1-82, arXiv:math/9906113, https://arxiv.org/abs/math/9906113
6. Saharon Shelah, "Logical dreams", Bulletin of the American Mathematical Society 40 (2003), 203-228, DOI 10.1090/S0273-0979-03-00981-9, https://doi.org/10.1090/s0273-0979-03-00981-9
7. Saharon Shelah, "More on the revised GCH and the black box", Annals of Pure and Applied Logic 140 (2006), 133-160, DOI 10.1016/j.apal.2005.09.013, https://doi.org/10.1016/j.apal.2005.09.013
8. Saharon Shelah, "Diamonds", Proceedings of the American Mathematical Society 138 (2010), 2151-2161, DOI 10.1090/S0002-9939-10-10254-8, https://doi.org/10.1090/s0002-9939-10-10254-8
9. Moti Gitik, "Extender-based forcings with overlapping extenders and negations of the Shelah Weak Hypothesis", Journal of Mathematical Logic 20 (2020), 2050013, DOI 10.1142/S0219061320500130, https://doi.org/10.1142/s0219061320500130
10. Moti Gitik and Saharon Shelah, "On some configurations related to the Shelah Weak Hypothesis", Archive for Mathematical Logic 40 (2001), 639-650, DOI 10.1007/s001530100076, https://doi.org/10.1007/s001530100076
11. Assaf Rinot, "A topological reflection principle equivalent to Shelah's Strong Hypothesis", Proceedings of the American Mathematical Society 136 (2008), 4413-4416, DOI 10.1090/S0002-9939-08-09411-2, https://doi.org/10.1090/S0002-9939-08-09411-2
12. Todd Eisworth, "A note on the Revised GCH", arXiv:2106.05369 (2021), https://arxiv.org/abs/2106.05369