APPROXIMATING DIAMOND PRINCIPLES ON PRODUCTS AT AN INACCESSIBLE CARDINAL

Omer Ben-Neria, Jing Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

We isolate the approximating diamond principles, which are consequences of the diamond principle at an inaccessible cardinal. We use these principles to find new methods for negating the diamond principle at large cardinals. Most notably, we demonstrate, using Gitik’s overlapping extenders forcing, a new method to get the consistency of the failure of the diamond principle at a large cardinal θ without changing cofinalities or adding fast clubs to θ. In addition, we show that the approximating diamond principles necessarily hold at a weakly compact cardinal. This result, combined with the fact that in all known models where the diamond principle fails the approximating diamond principles also fail at an inaccessible cardinal, exhibits essential combinatorial obstacles to make the diamond principle fail at a weakly compact cardinal.

Original languageAmerican English
Pages (from-to)5923-5948
Number of pages26
JournalTransactions of the American Mathematical Society
Volume376
Issue number8
DOIs
StatePublished - 2023

Bibliographical note

Publisher Copyright:
© 2023 American Mathematical Society.

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