Tree Property at Successors of Countable Limit Alephs

Definitions and mathematical statement

For an uncountable cardinal $\kappa$, a $\kappa$-tree is a partially ordered set $(T,<_{T})$ such that the strict predecessors of each node are well ordered by $<_{T}$, the tree has height $\kappa$, and every level has cardinality less than $\kappa$. More explicitly, if $\operatorname{ht}_{T}(t)$ is the order type of the predecessors of $t$, then \[ T_\xi=\{t\in T:\operatorname{ht}_{T}(t)=\xi\},\qquad 0<|T_\xi|<\kappa\quad(\xi<\kappa), \] and there are no nodes of height at least $\kappa$. A cofinal branch is a linearly ordered subset whose node heights are unbounded in $\kappa$. A $\kappa$-Aronszajn tree is a $\kappa$-tree with no cofinal branch. The tree property at $\kappa$, denoted here by $\operatorname{TP}(\kappa)$, asserts that every $\kappa$-tree has a cofinal branch.

Let $\omega_1$ be the first uncountable ordinal and let $\aleph_\beta$ denote the $\beta$-th infinite cardinal, all computed within the same model. Taking a limit ordinal to be nonzero and to have no immediate predecessor, the supplied problem asks whether the following theory is consistent: \[ \mathsf{ZFC}+\left[\forall\alpha<\omega_1\, \left(\alpha\text{ is a nonzero limit ordinal}\ \Longrightarrow\ \operatorname{TP}(\aleph_{\alpha+1})\right)\right]. \] Thus the tree properties are required simultaneously in one model for every countable nonzero limit ordinal. Interpreting consistency over $\mathsf{ZFC}$ is the usual ambient convention; the supplied record does not explicitly specify a base theory or a large-cardinal hypothesis for a relative consistency result. The nonzero convention for limit ordinals is also not explicit in the source record; including $0$ would add the separate requirement $\operatorname{TP}(\aleph_1)$.

Short English statement

Can there consistently be a single model of set theory in which, for every countable nonzero limit ordinal $\alpha$, every tree of height $\aleph_{\alpha+1}$ with levels of size less than $\aleph_{\alpha+1}$ has a cofinal branch? The base theory and the convention excluding zero from limit ordinals require confirmation.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003412$, title "Tree Property at Successors of Countable Limit Alephs," source problem code catalogue entry, collection "Oberwolfach Reports Open Problems." The record gives proposed year 2017 and names no proposer.
• Supplied report citation: Set Theory (2018), Oberwolfach Reports, pp. 527--589, DOI 10.4171/owr/2017/11. https://doi.org/10.4171/owr/2017/11. The supplied title year is 2018, whereas the DOI contains 2017; this discrepancy has not been resolved.
• Supplied literature source: "OWR Set Theory report." https://ems.press/content/serial-article-files/46674.
• Supplied literature source: "Diagonal supercompact Radin forcing." https://doi.org/10.1016/j.apal.2020.102828.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that no forcing construction currently establishes the simultaneous absence of Aronszajn trees in the stated pattern, while known tree-property results cover restricted patterns. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note reports that the official Oberwolfach report was checked directly and that the clean statement preserves the target while restoring definitions and removing source-dependent wording. These are supplied provenance claims; no independent source inspection is asserted here.