Approachable free subsets and fine structure derived scales

Dominik Adolf*, Omer Ben-Neria

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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
Issue number7
StatePublished - Jul 2024

Bibliographical note

Publisher Copyright:
© 2024 Elsevier B.V.


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


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