Approximating diamond principles on products at an inaccessible cardinal

Ben-Neria, Omer; Zhang, Jing

doi:10.1090/tran/8945

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

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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

Omer Ben-Neria, Diamonds, compactness, and measure sequences, J. Math. Log. 19 (2019), no. 1, 1950002, 20. MR 3960897, DOI 10.1142/S0219061319500028
Omer Ben-Neria and Jing Zhang. Compactness and guessing principles in the Radin extensions. Journal of Mathematical Logic, to appear.
Ari Meir Brodsky and Assaf Rinot, Distributive Aronszajn trees, Fund. Math. 245 (2019), no. 3, 217–291. MR 3914943, DOI 10.4064/fm542-4-2018
James Cummings, Woodin’s theorem on killing diamond via Radin forcing, Unpublished note.
James Cummings. private communication, 14 July 2013.
Mirna D amonja and Joel David Hamkins, Diamond (on the regulars) can fail at any strongly unfoldable cardinal, Ann. Pure Appl. Logic 144 (2006), no. 1-3, 83–95. MR 2279655, DOI 10.1016/j.apal.2006.05.001
Keith J. Devlin and Håvard Johnsbråten, The Souslin problem, Lecture Notes in Mathematics, Vol. 405, Springer-Verlag, Berlin-New York, 1974. MR 384542, DOI 10.1007/BFb0065979
Moti Gitik, Changing cofinalities and the nonstationary ideal, Israel J. Math. 56 (1986), no. 3, 280–314. MR 882254, DOI 10.1007/BF02782938
Moti Gitik, Prikry-type forcings, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1351–1447. MR 2768695, DOI 10.1007/978-1-4020-5764-9_{1}7
Moti Gitik, Blowing up the power of a singular cardinal of uncountable cofinality, J. Symb. Log. 84 (2019), no. 4, 1722–1743. MR 4045995, DOI 10.1017/jsl.2019.66
John Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), no. 3, 663–671. MR 485361, DOI 10.2307/2272043
Kai Hauser, Indescribable cardinals without diamonds, Arch. Math. Logic 31 (1992), no. 5, 373–383. MR 1164732, DOI 10.1007/BF01627508
R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR 309729, DOI 10.1016/0003-4843(72)90001-0
Ronald Jensen, Notes from Oberwolfach. Unpublished manuscript, 1991.
Ronald Jensen and Kenneth Kunen, Some combinatorial properties of L and V, Unpublished manuscript, 1969.
Jussi Ketonen, Some combinatorial principles, Trans. Amer. Math. Soc. 188 (1974), 387–394. MR 332481, DOI 10.1090/S0002-9947-1974-0332481-5
Carmi Merimovich, Extender-based Radin forcing, Trans. Amer. Math. Soc. 355 (2003), no. 5, 1729–1772. MR 1953523, DOI 10.1090/S0002-9947-03-03202-1
Justin Tatch Moore, Michael Hrušák, and Mirna D amonja, Parametrized $\diamondsuit$ principles, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2281–2306. MR 2048518, DOI 10.1090/S0002-9947-03-03446-9
Assaf Rinot, Jensen’s diamond principle and its relatives, Set theory and its applications, Contemp. Math., vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 125–156. MR 2777747, DOI 10.1090/conm/533/10506
Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912
Saharon Shelah, Not collapsing cardinals $\leq \kappa$ in $(<\kappa )$-support iterations, Israel J. Math. 136 (2003), 29–115. MR 1998104, DOI 10.1007/BF02807192
Saharon Shelah, Diamonds, Proc. Amer. Math. Soc. 138 (2010), no. 6, 2151–2161. MR 2596054, DOI 10.1090/S0002-9939-10-10254-8
Martin Zeman, $\diamondsuit$ at Mahlo cardinals, J. Symbolic Logic 65 (2000), no. 4, 1813–1822. MR 1812181, DOI 10.2307/2695076