Approachable free subsets and fine structure derived scales

Dominik Adolf*, Omer Ben-Neria

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal λ for which the set of Mitchell orders {o(μ)|μ<λ} is unbounded in λ. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.

Original languageAmerican English
Article number103428
JournalAnnals of Pure and Applied Logic
Volume175
Issue number7
DOIs
StatePublished - Jul 2024

Bibliographical note

Publisher Copyright:
© 2024 Elsevier B.V.

Keywords

  • Consistency and independence results
  • Inner models
  • Large cardinals
  • PCF theory

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