Total Failure of Approachability at Successors of Singulars of Countable Cofinality

TL;DR

The paper establishes the maximal failure of the approachability property at successors of countably cofinal singulars by constructing a class-forcing extension under which, for every singular $\delta$ with cf$(\delta)=\omega$ and each regular $\mu<\delta$, the set $E_{\mu}^{\delta^+}$ is not in the approachability ideal $I[\delta^+]$. It introduces a Namba-like forcing built from Laver-Ideal Property (LIP) ideals and reinforces previous work with a strengthened Prikry-type framework and Magidor iterations to collapse appropriate cardinals while preserving non-approachability. A key innovation is forcing $\delta^+$-level non-approachability without making $ heta^+$ $d$-approachable for any normal subadditive coloring $d$, achieved via LIP-based tail forcings and a careful projection/termspace analysis. The main result is a class-length, GCH-preserving iteration showing stationarily many non-approachable points of cofinality $\mu$ in $\delta^+$ for all countable cofinalities, answering longstanding questions of Mitchell and Foreman–Shelah.

Abstract

Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $δ$ of countable cofinality and every regular uncountable $μ<δ$ there are stationarily many non-approachable points of cofinality $μ$ in $δ^+$. This answers a question of Mitchell and provides a decisive answer to a question of Foreman and Shelah.

Theorems & Definitions (125)

• Theorem
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.7
• Theorem 1.8: Magidor
• proof
• Definition 1.9
• Theorem 1.10
• Lemma 1.11