Failure of Approachability at the Successor of the first Singular for any Cofinality
Hannes Jakob, Maxwell Levine
TL;DR
The paper resolves two long-standing questions about the combinatorics at the singular successor $aleph_{omega+1}$ by constructing models (via a sophisticated Easton-support Magidor iteration of Prikry-type forcings built from a Namba-style framework with Laver-type ideals) in which there are stationarily many points of cofinality $alph_n$ that are good but not approachable under $ extsf{GCH}$. The forcing framework collapses $aleph_{omega+1}$ to $alph_{n+1}$ while preserving cardinals and ensuring a weak approximation property, enabling precise control over scales and colorings that witness non-$d$-approachability for normal subadditive colorings $d$. The argument uses large-cardinal assumptions (supercompact embeddings) to lift and preserve non-approachability through the iteration, providing a robust separation between goodness and approachability and yielding corollaries about stationary sets of good-not-approachable points, including cofinalities $aleph_1$. Additionally, the paper demonstrates a model where a good scale coexists with failure of approachability, via a variant scale-adding poset that creates club-many good points. Overall, the results significantly advance the understanding of the interaction between scale structure, approachability, and goodness at the first singular successor.
Abstract
We solve two long-standing open problems regarding the combinatorics of $\aleph_{ω+1}$. We answer a question of Shelah by showing that it is consistent for any $n\geq 1$ that $\mathsf{GCH}$ holds and there is a stationary set of points of cofinality $\aleph_n$ which is not in the approachability ideal. As a corollary, we obtain a model where the notions of goodness and approachability are distinct for stationarily many points of cofinality $\aleph_1$, answering an open question of Cummings, Foreman, and Magidor.