Aronszajn-Tree Models from Nonstationary-Ideal Forcing

Definitions and mathematical statement

For a regular uncountable cardinal $\mu$, a $\mu$-Aronszajn tree has height $\mu$, levels of size less than $\mu$, and no branch of order type $\mu$. At $\mu=\nu^+$, it is special if its nodes are the union of $\nu$ antichains, equivalently if a map to $\nu$ is injective on each chain. Absence of all Aronszajn trees is the tree property and is stronger than absence of special Aronszajn trees.

The restored Question 6 asks for a particular forcing method, not merely a consistency conclusion. Start in a model with cardinals $\kappa<\lambda$, with $\kappa$ $\lambda$-supercompact and $\lambda$ weakly compact. In the standard embedding interpretation, the former gives an elementary embedding with critical point $\kappa$, image of $\kappa$ above $\lambda$, and target closed under $\lambda$-sequences; weak compactness has its usual large-cardinal meaning. Can an iteration of the report's nonstationary-ideal forcing method reprove the cited Abraham conclusion, producing a generic extension $V[G]$ in which \[ \text{there are no }\aleph_2^{V[G]}\text{-Aronszajn trees and no } \aleph_3^{V[G]}\text{-Aronszajn trees}? \] The cardinal symbols in the target refer to the extension. The source does not specify the collapse bookkeeping identifying ground-model cardinals with them.

As a weaker fallback, it asks whether that method can at least construct a model with no special Aronszajn trees on $\aleph_2$ or $\aleph_3$. The source's "or" is retained: the precise requirement of one target versus both simultaneously needs confirmation. This fallback must not be strengthened to the full tree property without proof.

The particular nonstationary-ideal forcing posets, iteration support, preservation arguments and any extra hypotheses are not supplied in the extract. A construction by another technique does not answer the method-specific request. The literature assessment checked on 21 August 2026 reports consistency models with no special trees at many successive regular cardinals, including $\aleph_2$, but does not claim the requested method reproves Abraham's theorem from exactly the displayed assumptions. These are dated supplied partial-progress claims requiring independent verification.

Short English statement

Can the report's nonstationary-ideal forcing iteration recover Abraham's simultaneous tree-property result at $\aleph_2$ and $\aleph_3$ under the stated large cardinals? The weaker fallback excludes only special trees, and existing constructions by other methods do not settle this request.

Sources

• Feinstrukturtheorie und Innere Modelle (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/20, pp. 1215--1244, Question 6 according to supplied verification, https://doi.org/10.4171/owr/2006/20; proposed year supplied as 2006. The restored theorem is attributed to Abraham; no separate full Abraham citation is supplied.
• Literature assessment checked 21 August 2026: original report, https://doi.org/10.4171/owr/2006/20; Unger, Successive failures of approachability, https://arxiv.org/abs/1702.05062.
• Local UnsolvedMath registry, permanent ID $30000472$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. Supplied verification expands the referenced theorem and retains the method request.