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 language | English |
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Article number | 2250024 |
Pages (from-to) | 2250024:1-2250024:22 |
Number of pages | 22 |
Journal | Journal of Mathematical Logic |
Volume | 23 |
Issue number | 2 |
DOIs | |
State | Published - 1 Aug 2023 |
Bibliographical note
Publisher Copyright:© 2023 World Scientific Publishing Company.
Keywords
- Radin forcing
- amenable C -sequence
- diamond
- weak compact