Optimal hypotheses for Easton's theorem at measurable and strong cardinals

Easton's theorem (1970) shows that, assuming GCH, the values of the continuum function on regular cardinals can be forced to follow any Easton function: a monotone class function F on the regular cardinals with cf(F(α)) > α. Friedman and Honzik (2008) extended Easton's theorem into the large-cardinal realm: a class-length Easton-support iteration P^F, mixing Cohen forcing with generalized Sacks forcing, realizes F while preserving the measurability of any cardinal κ that is a closure point of F (F(α) < κ for all regular α < κ) and is F(κ)-hypermeasurable via an embedding j satisfying the compatibility condition j(F)(κ) ≥ F(κ), and preserves all strong cardinals when F is locally definable. The problem studied here is whether these hypotheses are optimal. Three questions remain open: (1) the exact minimal consistency strength — measured by the Mitchell order o(κ) — needed to realize a prescribed Easton function with 2^κ = F(κ) > κ^+ while keeping κ measurable; (2) whether the compatibility condition j(F)(κ) ≥ F(κ) is necessary for lifting, or can be replaced by a weaker condition on F and j; (3) whether local definability of F can be removed when preserving strong cardinals. Later literature resolved important special cases — optimal-hypothesis realization of F(κ) = κ^{+n} for finite n ≥ 2 under a club-many-failures condition (Friedman–Honzik 2012), removal of local definability for Woodin cardinals and for Ramsey and strongly Ramsey cardinals (Cody 2013; Cody–Gitman 2015), preservation of extendible-type cardinals under any Δ2-definable behavior of the power-set function (Bagaria–Poveda 2023), and necessity of analogous compatibility conditions for supercompactness (Cody–Magidor 2014; Cody–Friedman–Honzik 2014) — but the sharp, general Easton theorem at measurable and strong cardinals, with upper and lower bounds matching, is still missing.

Background

By the continuum function one means the map $\kappa\mapsto 2^\kappa$ on the infinite cardinals; GCH (the generalized continuum hypothesis) asserts $2^\kappa=\kappa^+$ for every infinite $\kappa$, where $\kappa^+$ is the next larger cardinal. A cardinal is regular if it equals its own cofinality, the least order type of an unbounded subset. Easton's theorem [Easton 1970] shows that, over a model of GCH, the continuum function on the class of regular cardinals can be set by forcing to agree with any Easton function: a class function $F$ from regular cardinals to cardinals satisfying (i) monotonicity, $\alpha\le\beta\Rightarrow F(\alpha)\le F(\beta)$, and (ii) König's constraint $\operatorname{cf}(F(\alpha))>\alpha$. These two conditions are the only constraints ZFC imposes on the continuum function at regular cardinals; Easton's forcing is an Easton-support iteration (supports bounded below each regular stage) of Cohen forcings $\operatorname{Add}(\alpha,F(\alpha))$, each adding $F(\alpha)$-many subsets of $\alpha$.

Large cardinals change the picture. A cardinal $\kappa$ is measurable if there is an elementary embedding $j:V\to M$ of the universe into a transitive class $M$ with critical point $\kappa$ (the least ordinal moved by $j$); equivalently, $\kappa$ carries a $\kappa$-complete nonprincipal ultrafilter, a normal measure. Reflection constrains the continuum function at measurables: Scott's classical reflection theorem (recalled in [Cody 2012, dissertation]) shows that if $2^\kappa>\kappa^+$ at a measurable $\kappa$, then the set $\{\delta<\kappa:2^\delta>\delta^+\}$ must belong to some normal measure on $\kappa$, so a GCH failure at $\kappa$ is always anticipated below $\kappa$. The consistency strength of such failures is calibrated by the Mitchell order $o(\kappa)$, which ranks how many coherent layers of normal measures (or extenders, for higher values) exist on $\kappa$: the least GCH failure at a measurable cardinal, $2^\kappa=\kappa^{++}$, has consistency strength exactly $o(\kappa)=\kappa^{++}$ [Gitik 1993] — the lower bound via the core model for sequences of measures [Mitchell 1984], and the upper bound by forcing from $o(\kappa)=\kappa^{++}$; prescribed larger values of $2^\kappa$ require correspondingly greater strength. Here $H(\theta)$ denotes the collection of sets of hereditary size less than $\theta$.

Friedman and Honzik [2008] proved a large-cardinal Easton theorem. Say $\kappa$ is $F(\kappa)$-hypermeasurable if there is $j:V\to M$ with critical point $\kappa$ such that $F(\kappa)<j(\kappa)$ and $H(F(\kappa))^V\subseteq M$; $\kappa$ is strong if for every $\theta$ there is an embedding $j:V\to M$ with critical point $\kappa$ such that $j(\kappa)>\theta$ and $H(\theta)^V\subseteq M$; and $F$ is locally definable if some fixed pair of formulas defines $F\restriction\gamma$ over $H(\gamma)$ whenever $H(\gamma)$ satisfies a fixed sentence (a condition introduced by Menas [1976] for supercompact cardinals). Their forcing $\mathbb{P}^F$ is a class-length Easton-support iteration mixing Cohen forcing at successors with generalized Sacks forcing (a perfect-tree forcing adding many subsets at inaccessible closure points, whose fusion structure makes embeddings liftable) at inaccessibles. They showed: if $\kappa$ is a closure point of $F$ ($F[\kappa]\subseteq\kappa$) and is $F(\kappa)$-hypermeasurable via $j$ with the compatibility condition $j(F)(\kappa)\ge F(\kappa)$, where $j(F)$ is the pointwise image of $F$ under $j$, then $\mathbb{P}^F$ realizes $F$ while preserving the measurability of $\kappa$; and if moreover $F$ is locally definable, $\mathbb{P}^F$ realizes $F$ while preserving all strong cardinals. The lifting step uses Silver's lifting criterion [Silver 1971]: a generic filter for $j(\mathbb{P}^F)$ over $M$ containing the pointwise image $j[G]$ yields an extension of $j$ to the forcing extension.

The open problem is the optimality of these hypotheses. The $H(F(\kappa))$-closure hypothesis corresponds, under GCH, to Mitchell-order strength slightly above the core-model lower bounds (Honzík's survey calls it 'almost optimal'), and the compatibility and local-definability conditions may be artifacts of the lifting technique rather than genuine requirements; in the adjacent supercompact setting, analogous compatibility conditions have been proved necessary [Cody–Magidor 2014; Cody–Friedman–Honzik 2014].

Problem Statement

Work over ZFC + GCH, the standard background for all results in this line. Let $F$ be an Easton function, i.e. a class function from the regular cardinals to the cardinals with $\alpha\le\beta\Rightarrow F(\alpha)\le F(\beta)$ and $\operatorname{cf}(F(\alpha))>\alpha$. The objective is a sharp Easton theorem at the measurable and strong cardinals of the universe: determine, with hypotheses that are provably no stronger than necessary, exactly when $F$ can be realized ($2^\alpha=F(\alpha)$ for every regular $\alpha$, by cofinality-preserving forcing) while preserving these large cardinals. Three interlocking questions concerning the optimality of the hypotheses of the main theorems of [Friedman–Honzik 2008] remain open; the optimality program was pursued in [Friedman–Honzik 2012] and the subsequent literature.

(1) Minimal strength for measurability. [Friedman–Honzik 2008] realizes $F$ with $\kappa$ measurable from an embedding $j:V\to M$ with $H(F(\kappa))^V\subseteq M$ and $F(\kappa)<j(\kappa)$ — strength slightly above the known lower bounds [Mitchell 1984; Gitik 1993]. Determine the exact consistency strength, in terms of the Mitchell order $o(\kappa)$ or the corresponding extender hypotheses, of '$\kappa$ is measurable and $2^\kappa=F(\kappa)$ with $F(\kappa)>\kappa^+$' — jointly with a prescribed Easton behavior of $F$ below $\kappa$. Known: the answer is optimal for $F(\kappa)=\kappa^{+n}$, finite $n\ge 2$, under the additional assumption that $\{\alpha<\kappa$ regular$: F(\alpha)\ge\alpha^{++}\}$ contains all regulars of a club [Friedman–Honzik 2012]. Still open: (a) the case $F(\kappa)=\kappa^{+\beta}$ for infinite $\beta\ge\omega$ (Question 1 of [Friedman–Honzik 2012]) — the preprint [Ben Shalom 2017] treats only the uniform successor-exponent patterns, leaving the general case open; (b) whether the club-many-failures condition can be weakened — the classical reflection theorem makes stationarity of that set necessary, so the exact boundary is undetermined; (c) whether, for values of $F(\kappa)$ beyond the finite-gap range, the $H(F(\kappa))$-closure hypothesis can be matched exactly to the lower bound.

(2) Compatibility condition. Is the condition $j(F)(\kappa)\ge F(\kappa)$ — where $j(F)$ is the pointwise image of the Easton function under the witnessing embedding — necessary for the lifting of $j$ through the Easton forcing, or can it be replaced by a strictly weaker relation between $F$ and $j$ that still permits lifting? Note that any weaker replacement condition must remain dependent on the witnessing embedding $j$: the apparent lack of uniformity of the condition is unavoidable [Honzík 2016]. (For the supercompact interval analogue, the corresponding conditions $|j(F)(\gamma)|=F(\gamma)$ and $j(\kappa)>F(\lambda)$ are known to be necessary [Cody–Friedman–Honzik 2014; Cody–Magidor 2014]; the measurable case has only measure-specific obstructions [Honzík 2016].)

(3) Local definability for strong cardinals. Can the local definability hypothesis on $F$ be removed, or weakened, so that an Easton-style forcing realizes arbitrary $F$ while preserving all strong cardinals? It has been removed for Woodin cardinals [Cody 2013], Ramsey and strongly Ramsey cardinals [Cody–Gitman 2015], and (for $\Delta_2$-definable power-set behavior) $C^{(n)}$-extendible cardinals [Bagaria–Poveda 2023], but it remains open for strong cardinals, where every located statement still assumes it.

Questions (1)–(3) are three individually open questions; progress on any one of them is a meaningful contribution, in either direction — a strengthening of the known sufficient hypotheses toward the core-model lower bounds, a proof that one of the residual assumptions (such as club-many failures, the compatibility condition, or some form of local definability) is essential, or a removal of such an assumption. The overarching scientific objective — the sharp large-cardinal Easton theorem at measurable and strong cardinals, closing the gaps between the current sufficient hypotheses and the known necessary conditions — remains open: no complete generalization of the Easton theorem has yet been formulated [Honzík 2016].

References

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