Compactness and guessing principles in the Radin extensions

Omer Ben-Neria*, Jing Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on κ, if κ is weakly compact, then (κ) holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from [O. B. -Neria, Diamonds, compactness, and measure sequences, J. Math. Log. 19(1) (2019) 1950002].

Original languageAmerican English
Article number2250024
Pages (from-to)2250024:1-2250024:22
Number of pages22
JournalJournal of Mathematical Logic
Volume23
Issue number2
DOIs
StatePublished - 1 Aug 2023

Bibliographical note

Publisher Copyright:
© 2023 World Scientific Publishing Company.

Keywords

  • Radin forcing
  • amenable C -sequence
  • diamond
  • weak compact

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