Compactness and guessing principles in the Radin extensions

Omer Ben-Neria*, Jing Zhang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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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