The residual program for Shelah's approachability ideal at successors of singular cardinals: uniform non-approachability, global failure of SCH and approachability, Woodin's AP-versus-SCH question, and the Tree Property at every successor of a singular (Jakob-Poveda, Questions 5.1-5.4)

For a singular cardinal $\kappa$, Shelah's approachability ideal $I[\kappa^+]$ and the approachability property $\mathrm{AP}_\kappa$ (the assertion that $I[\kappa^+]$ is improper) measure how much of $\kappa^+$ can be enumerated by small initial segments, and sit at the junction of stationary-set preservation, weak square, and the Tree Property. Jakob and Poveda (arXiv:2508.04374, 2025) proved from GCH plus one supercompact cardinal that for any prescribed singular $\aleph_\gamma$ and regular $\mu\in(\mathrm{cf}(\gamma),\aleph_\gamma)$ there is a forcing extension with $\aleph_{\gamma+1}\cap\mathrm{cof}(\mu)\notin I[\aleph_{\gamma+1}]$ (settling a 1980s question of Shelah), and from a proper class of supercompact cardinals that $\mathrm{AP}_\kappa$ fails at every singular $\kappa$ simultaneously. The closing section of their paper poses four open questions that jointly define the surviving core of the approachability program: (5.1) a model in which, below one singular $\lambda$ of uncountable cofinality, every regular $\mu\in(\mathrm{cf}(\lambda),\lambda)$ carries a stationary subset of $\lambda^+\cap\mathrm{cof}(\mu)$ outside $I[\lambda^+]$; (5.2) the consistency, from a proper class of supercompacts, of the Singular Cardinals Hypothesis failing on a club class while $\mathrm{AP}_\kappa$ fails at every singular $\kappa$; (5.3) Woodin's 1980s question whether $\mathrm{AP}_\kappa$ implies the SCH at a singular strong limit $\kappa$ in ZFC; and (5.4) the Tree Property at $\kappa^+$ for every singular $\kappa$. All four questions remain open.

Background

A cardinal is singular if its cofinality $\mathrm{cf}(\kappa)$, the least size of an unbounded subset of $\kappa$, is smaller than $\kappa$ (the first example is $\aleph_\omega=\sup_n\aleph_n$). Unlike regular cardinals, where Easton showed ZFC imposes essentially no constraint on the continuum function $\kappa\mapsto 2^\kappa$, singular cardinals obey deep ZFC theorems (Silver, Magidor, Shelah's PCF theory). The Singular Cardinals Hypothesis (SCH) asserts that every singular strong limit cardinal $\kappa$ (strong limit meaning $2^\mu<\kappa$ for all $\mu<\kappa$) satisfies $\kappa^{\mathrm{cf}(\kappa)}=\kappa^+$, equivalently $2^\kappa=\kappa^+$; where SCH fails is the subject of the singular cardinals problem, and all known violations at small singulars require large cardinal axioms.

Shelah introduced the approachability ideal in his 1979 study of successors of singular cardinals. For a regular $\theta$, a sequence $\vec a=\langle a_\alpha\mid\alpha<\theta\rangle$ of bounded subsets of $\theta$ and a limit $\alpha<\theta$, one says that $\alpha$ is approachable with respect to $\vec a$ if there is an unbounded $A\subseteq\alpha$ with $\mathrm{otp}(A)=\mathrm{cf}(\alpha)$ such that every initial segment $A\cap\beta$ ($\beta<\alpha$) occurs as $a_\gamma$ for some $\gamma<\alpha$. The ideal $I[\theta]$ consists of those $S\subseteq\theta$ for which some club $D\subseteq\theta$ and some such $\vec a$ make every point of $S\cap D$ approachable; $I[\theta]$ is a (possibly improper) normal ideal containing all nonstationary sets. Here $\theta\cap\mathrm{cof}(\mu)$ denotes the ordinals below $\theta$ of cofinality $\mu$, and a club is a closed unbounded subset. The approachability property $\mathrm{AP}_\kappa$ for a singular $\kappa$ asserts that $I[\kappa^+]$ is improper ($\kappa^+\in I[\kappa^+]$). The ideal's significance is that it calibrates exactly which stationary sets on $\kappa^+$ ZFC forces to be approachable and which are malleable by forcing: for singular $\kappa$, ZFC proves $\kappa^+\cap\mathrm{cof}(\le\mathrm{cf}(\kappa))\in I[\kappa^+]$, but the points of cofinality strictly between $\mathrm{cf}(\kappa)$ and $\kappa$ are not so constrained.

Shelah (1979), from a supercompact cardinal, produced a model with $\aleph_{\omega+1}\cap\mathrm{cof}(\omega_1)\notin I[\aleph_{\omega+1}]$, his argument being limited to cofinality $\omega_1$; the $\omega_2$-cofinality instance was posed as Question 8.3 of Foreman's 2005 problem list and remained open until it was answered by Jakob-Levine (2025). Jakob (2025) then obtained the total failure $\mu^+\cap\mathrm{cof}(\gamma)\notin I[\mu^+]$ simultaneously for all regular $\gamma\in(\aleph_0,\mu)$ and all singular $\mu$ of countable cofinality, using Namba-style forcings available only at countable cofinality. Jakob-Poveda (2025) broke the countable-cofinality barrier with a new poset $\mathcal P(\vec\kappa,\mathcal I,\vec B)$, an ideal-based version of a Magidor product of one-point Prikry forcings (Prikry-type forcings are a family of posets that alter the cofinalities of large cardinals without adding bounded subsets and admit a well-developed iteration theory), proving: (Theorem A) from GCH plus one supercompact cardinal, for any singular $\aleph_\gamma$ and regular $\mu\in(\mathrm{cf}(\gamma),\aleph_\gamma)$ there is a model with $\aleph_{\gamma+1}\cap\mathrm{cof}(\mu)\notin I[\aleph_{\gamma+1}]$; and (Theorem B) from GCH plus a proper class of supercompact cardinals there is a model where $\mathrm{AP}_\kappa$ fails at every singular $\kappa$, with unboundedly many regular $\delta<\kappa$ witnessing $\kappa^+\cap\mathrm{cof}(\delta)\notin I[\kappa^+]$.

Two further principles connect $I[\kappa^+]$ to the rest of singular cardinal combinatorics. Jensen's weak square $\square^*_\kappa$ (a sequence $\langle C_\alpha\mid\alpha<\kappa^+\rangle$ with $C_\alpha$ club in $\alpha$, of order type at most $\kappa$, whose initial segments cohere) implies $\mathrm{AP}_\kappa$, and by a classical theorem of Jensen (1972) recorded in the source paper it is equivalent to the existence of a special $\kappa^+$-Aronszajn tree, that is, a $\kappa^+$-tree admitting a function $f:T\to\kappa$ with $x<_T y\Rightarrow f(x)\neq f(y)$; such trees remain branchless in every outer model preserving $\kappa^+$, so they are absolute witnesses to the failure of the Tree Property. The Tree Property at a regular $\theta$ (every $\theta$-tree, i.e., tree of height $\theta$ with levels of size $<\theta$, has a cofinal branch) is the compactness principle behind Foreman and Magidor's 1980s question of whether some regular $\kappa\ge\aleph_2$ must fail it; getting it at the successor of every singular cardinal is the recognized principal obstacle. On the arithmetic side, Gitik and Sharon (2008) produced from a supercompact a model with $\neg\mathrm{SCH}+\neg\mathrm{AP}$ plus a very good scale at a singular $\kappa$, answering a consistency question of Woodin from the 1980s about $\neg\mathrm{SCH}+\neg\square^*$; Ben-Neria, Lambie-Hanson and Unger (2020) introduced diagonal supercompact Radin forcing and produced a model with many singular cardinals at which both the SCH and weak square fail, and Gitik (2021) answered their question with a model in which AP and SCH fail on a club class. The Tree Property lineage runs from Magidor and Shelah (1996), through Neeman (2009), who first combined a Tree Property at $\lambda^+$ with failure of SCH at $\lambda$, Neeman (2014) up to $\aleph_{\omega+1}$, Golshani and Hayut (2018) on countable segments of successors, Cummings, Hayut, Magidor, Neeman, Sinapova and Unger (2021) at both immediate successors of a singular and, in a 2020 companion paper, with the stronger ineffable tree property together with failure of the SCH, Adkisson (2025, 2026) at successors of many cofinalities and countable segments, and Cummings et al. (2025) on long intervals of regular cardinals overlapping a strong limit.

Against this backdrop, Jakob-Poveda's Section 5 records what their methods cannot reach. Their poset has stronger requirements than the Namba-style one it replaces, so the collapses must be separated and only a weak simultaneity property (SLIP, the Simultaneous Laver Ideal Property of their Definition 3.6) is available, blocking all cofinalities at once (5.1). Combining their global AP-failure iteration with global SCH failure is open (Question 5.2). And Theorem B, by failing $\mathrm{AP}_\kappa$ (hence $\square^*_\kappa$, hence all special $\kappa^+$-Aronszajn trees) at every singular, removes only the special trees, not every $\kappa^+$-Aronszajn tree, leaving the full Tree Property open (5.4).

Problem Statement

The problem is the suite of four open questions with which Jakob-Poveda (arXiv:2508.04374, Section 5) close their resolution of Shelah's approachability question. They share one scientific objective: to determine the extent to which the approachability ideal at successors of singular cardinals can be controlled, and how its failure interacts with cardinal arithmetic (the SCH) and with compactness (the Tree Property) at class scale. The four questions, in the source's own order and formulation, are:

(5.1) Uniform simultaneous non-approachability. Suppose $\lambda$ is a singular cardinal of uncountable cofinality. Is there a model of ZFC plus "For every regular $\mu\in(\mathrm{cf}(\lambda),\lambda)$ there is a stationary subset of $\lambda^+\cap\mathrm{cof}(\mu)$ which is not in $I[\lambda^+]$"? (Theorem A achieves any single prescribed $\mu$; the source's methods require the collapses to be separated and cannot reach all cofinalities at once.)

(5.2) Global SCH failure with global AP failure. Assuming a proper class of supercompact cardinals, is it consistent with ZFC that the SCH fails on a club class (of singular cardinals) and, simultaneously, $\mathrm{AP}_\kappa$ fails at every singular cardinal $\kappa$?

(5.3) Woodin's question (1980s). Let $\kappa$ be a singular strong limit cardinal. Does the approachability property $\mathrm{AP}_\kappa$ imply the SCH at $\kappa$? (A ZFC implication question.)

(5.4) Tree Property at all successors of singulars. Assuming appropriate large cardinals, is it possible to have a model of ZFC in which the Tree Property holds at $\kappa^+$ for every singular cardinal $\kappa$? (Theorem B already eliminates all special $\kappa^+$-Aronszajn trees, but the full Tree Property requires eliminating every $\kappa^+$-Aronszajn tree.)

Each of 5.1-5.4 is an independently stated and independently refutable target: settling any one of them resolves that question and leaves the other three open, and the four are not mutually conditioning.

References

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