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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Approximating diamond principles on products at an inaccessible cardinal
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by Omer Ben-Neria and Jing Zhang
Trans. Amer. Math. Soc. 376 (2023), 5923-5948
DOI: https://doi.org/10.1090/tran/8945
Published electronically: May 19, 2023

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 $\theta$ without changing cofinalities or adding fast clubs to $\theta$. 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.
References
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Bibliographic Information
  • Omer Ben-Neria
  • Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, Israel
  • MR Author ID: 1061476
  • ORCID: 0000-0003-1277-9384
  • Email: omer.bn@mail.huji.ac.il
  • Jing Zhang
  • Affiliation: Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Room 6290, Toronto, Ontario, Canada M5S 2E
  • Email: jingzhan@alumni.cmu.edu
  • Received by editor(s): September 2, 2022
  • Received by editor(s) in revised form: February 27, 2023
  • Published electronically: May 19, 2023
  • Additional Notes: 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).
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 5923-5948
  • MSC (2020): Primary 03E02, 03E35, 03E55
  • DOI: https://doi.org/10.1090/tran/8945
  • MathSciNet review: 4630763