Class-many strong limit singular cardinals with $2^{\aleph_\delta}<\aleph_{(|\delta|^{+3})}$ in ZFC (Garti's Conjecture 3.4 on weak diamond and pcf theory)

Garti's preprint “Weak diamond and pcf theory” (arXiv:2405.03142, 2024) isolates a precise open question at the frontier of Shelah's pcf theory: whether ZFC alone proves that there is a proper class of strong limit singular cardinals $\aleph_\delta$ with $2^{\aleph_\delta}<\aleph_{(|\delta|^{+3})}$ (Conjecture 3.4). The paper proves this conditionally — from instances of the Devlin–Shelah weak diamond, for example under the hypothesis that every limit cardinal is a strong limit cardinal — improving by one successor, at class-many singular cardinals, Shelah's celebrated 1994 bound $2^{\aleph_\delta}<\aleph_{|\delta|^{+4}}$ for strong limit $\aleph_\delta$. The problem — prove or refute the class-many +3 power-set bound in ZFC — remains open.

Background

Cardinal arithmetic studies the power-set function $\kappa\mapsto 2^\kappa$ on infinite cardinals. Easton's theorem (1970) determines exactly which behaviors of $2^\kappa$ on regular cardinals are consistent with ZFC, so the deep open questions concern singular cardinals. A cardinal $\lambda$ is singular if its cofinality $\operatorname{cf}(\lambda)$ — the least size of an unbounded subset of $\lambda$ — is smaller than $\lambda$; $\lambda$ is a strong limit if $2^\theta<\lambda$ for every $\theta<\lambda$, a hypothesis that removes the trivial way a singular cardinal inherits a large power set from below. Writing $\aleph_\delta$ for the $\delta$-th infinite cardinal ($\delta$ an ordinal) and $|\delta|$ for the cardinality of the ordinal $\delta$, the singular cardinals hypothesis (SCH) asserts $2^\lambda=\lambda^+$ for every strong limit singular $\lambda$. Silver proved that SCH cannot first fail at a singular of uncountable cofinality, and Magidor proved that SCH can consistently fail already at $\aleph_\omega$; consequently, the exact bounds on $2^\lambda$ at singular cardinals provable in ZFC alone became one of the central problems of set theory.

Galvin and Hajnal (1975) obtained the first general ZFC bounds on powers of singulars in terms of the aleph index. The decisive tool is Shelah's pcf theory: for a set $\mathfrak a$ of regular cardinals, $pcf(\mathfrak a)$ is the set of cofinalities of ultraproducts $\prod\mathfrak a/D$ over ultrafilters $D$ on $\mathfrak a$; for a strong limit singular $\lambda$ that is not a fixed point of the aleph function (i.e., $\lambda=\aleph_\delta$ with $\aleph_\delta\neq\delta$), $2^\lambda=\max pcf(\mathfrak a)$ for a suitable progressive end-segment $\mathfrak a$ of the regular cardinals below $\lambda$, so bounds on $|pcf(\mathfrak a)|$ become bounds on $2^\lambda$. A cornerstone of Shelah's book “Cardinal Arithmetic” (1994) is: if $\aleph_\delta$ is a strong limit cardinal and $\delta<\aleph_\delta$, then

$$2^{\aleph_\delta}<\aleph_{|\delta|^{+4}},$$

where $\kappa^{+4}$ is the fourth successor cardinal of $\kappa$; at $\delta=\omega$ this is the celebrated $2^{\aleph_\omega}<\aleph_{\omega_4}$ for strong limit $\aleph_\omega$. No improvement of the +4 has been proved in ZFC since, and reducing it — even to +3, even for just a proper class of singular cardinals rather than all of them — is a recognized open frontier; the paper formalizes the class-many +3 statement as an explicit conjecture.

Garti's “Weak diamond and pcf theory” (2024) attacks this via two classical tools. The first is the Devlin–Shelah weak diamond $\Phi_\kappa$: for every coloring $c:{}^{<\kappa}2\to2$ there is a $g:\kappa\to2$ such that for every $f:\kappa\to2$ the set $\{\alpha<\kappa: c(f{\restriction}\alpha)=g(\alpha)\}$ is stationary in $\kappa$ (it meets every closed unbounded subset of $\kappa$). Unlike the full diamond prediction principle, weak diamond is equivalent at successors to the arithmetic statement $2^\kappa<2^{\kappa^+}$, and — by Benhamou–Garti–Poveda (2023), building on Garti's earlier link between weak diamond and Galvin's property, a covering property of filters introduced by Galvin — it holds, in ZFC alone, at a proper class of cardinals. The second tool is club guessing: a sequence $(C_\alpha)_{\alpha\in S^\lambda_\kappa}$, where $S^\lambda_\kappa=\{\alpha<\lambda:\operatorname{cf}(\alpha)=\kappa\}$, of closed unbounded sets $C_\alpha\subseteq\alpha$ that guesses every club $E\subseteq\lambda$ in the sense that $C_\alpha\subseteq E$ for some $\alpha$. Shelah's +4 bound flows from club guessing at $S^{\eta^{++}}_{\eta}$; Garti's Theorem 3.1 proves the matching fact at the next cofinality: club guessing at $S^{\eta^{++}}_{\eta^+}$ yields $|pcf(\mathfrak a)|<|\mathfrak a|^{+3}$ for every progressive interval $\mathfrak a$ of regular cardinals (that is, $|\mathfrak a|<\min\mathfrak a$). Corollary 3.2 converts a weak-diamond instance into the local cardinal-arithmetic bound: if $\lambda=\aleph_\delta$ is strong limit singular and not a fixed point of the aleph function ($\aleph_\delta\neq\delta$), and $2^{\eta^+}<2^{\eta^{++}}$ at the relevant scale $\eta$, then $2^\lambda<\aleph_{(|\delta|^{+3})}$. Theorem 3.3 then shows that the arithmetic hypothesis ‘every limit cardinal is a strong limit cardinal’ supplies weak-diamond instances at unboundedly many points, and hence a proper class of strong limit singulars with the +3 bound. The paper closes by describing the pathological regimes in which the local implication can fail — long intervals of regular cardinals carrying a constant value of the power set, or values of $2^{\eta^+}$ that are weakly but not strongly inaccessible — and states Conjecture 3.4: no arithmetic hypothesis beyond ZFC should be needed.

Problem Statement

Prove or refute Conjecture 3.4 of Garti's “Weak diamond and pcf theory” (arXiv:2405.03142), stated in the paper's own notation:

In every model of ZFC there is a class of strong limit singular cardinals of the form $\aleph_\delta$ for which

$$2^{\aleph_\delta}<\aleph_{(|\delta|^{+3})}.$$

Throughout, $\aleph_\delta$ is the $\delta$-th infinite cardinal ($\delta$ an ordinal), $|\delta|$ is the cardinality of the ordinal $\delta$, and $\kappa^{+3}$ is the third successor cardinal of $\kappa$; a cardinal $\lambda$ is singular if its cofinality (the least size of an unbounded subset of $\lambda$) is smaller than $\lambda$, and a strong limit if $2^\theta<\lambda$ for every cardinal $\theta<\lambda$. The phrase ‘there is a class of’ is read as ‘there are proper-class many’, i.e., unboundedly many indices. Equivalently: prove in ZFC, or refute, the statement that for every ordinal $\alpha$ there exists an ordinal $\delta>\alpha$ such that $\aleph_\delta$ is a strong limit singular cardinal and $2^{\aleph_\delta}<\aleph_{(|\delta|^{+3})}$.

References

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