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.
Bibliographical noteFunding Information:
Received by the editors September 2, 2022, and, in revised form, February 27, 2023. 2020 Mathematics Subject Classification. Primary 03E02, 03E35, 03E55. The first author was partially supported by the Israel Science Foundation (Grant 1832/19). The second author was supported by the European Research Council (grant agreement ERC-2018-StG 802756).
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