The ineffable tree property and failure of the singular cardinals hypothesis

Cummings, James; Hayut, Yair; Magidor, Menachem; Neeman, Itay; Sinapova, Dima; Unger, Spencer

doi:10.1090/tran/8110

The ineffable tree property and failure of the singular cardinals hypothesis
HTML articles powered by AMS MathViewer

by James Cummings, Yair Hayut, Menachem Magidor, Itay Neeman, Dima Sinapova and Spencer Unger PDF

Trans. Amer. Math. Soc. 373 (2020), 5937-5955 Request permission

Abstract:

ITP is a combinatorial principle that is a strengthening of the tree property. For an inaccessible cardinal $\kappa$, ITP at $\kappa$ holds if and only if $\kappa$ is supercompact. And just like the tree property, it can be forced to hold at accessible cardinals. A broad project is obtaining ITP at many cardinals simultaneously. Past a singular cardinal, this requires failure of SCH. We prove that from large cardinals, it is consistent to have failure of SCH at $\kappa$ together with ITP $\kappa ^+$. Then we bring down the result to $\kappa =\aleph _{\omega ^2}$.

References

James Cummings and Matthew Foreman, The tree property, Adv. Math. 133 (1998), no. 1, 1–32. MR 1492784, DOI 10.1006/aima.1997.1680
Laura Fontanella, Strong tree properties for small cardinals, J. Symbolic Logic 78 (2013), no. 1, 317–333. MR 3087079, DOI 10.2178/jsl.7801220
Moti Gitik and Assaf Sharon, On SCH and the approachability property, Proc. Amer. Math. Soc. 136 (2008), no. 1, 311–320. MR 2350418, DOI 10.1090/S0002-9939-07-08716-3
Dima Sinapova and Sherwood Hachtman, The super tree property at the successor of a singular, to appear.
M. Magidor, Combinatorial characterization of supercompact cardinals, Proc. Amer. Math. Soc. 42 (1974), 279–285. MR 327518, DOI 10.1090/S0002-9939-1974-0327518-9
Itay Neeman, The tree property up to $\aleph _{\omega +1}$, J. Symb. Log. 79 (2014), no. 2, 429–459. MR 3224975, DOI 10.1017/jsl.2013.25
Itay Neeman, Aronszajn trees and failure of the singular cardinal hypothesis, J. Math. Log. 9 (2009), no. 1, 139–157. MR 2665784, DOI 10.1142/S021906130900080X
Dima Sinapova, The tree property and the failure of the singular cardinal hypothesis at $\aleph _{\omega ^2}$, J. Symbolic Logic 77 (2012), no. 3, 934–946. MR 2987144, DOI 10.2178/jsl/1344862168
Spencer Unger, A model of Cummings and Foreman revisited, Ann. Pure Appl. Logic 165 (2014), no. 12, 1813–1831. MR 3256740, DOI 10.1016/j.apal.2014.07.002
Matteo Viale, A family of covering properties, Math. Res. Lett. 15 (2008), no. 2, 221–238. MR 2385636, DOI 10.4310/MRL.2008.v15.n2.a2
Matteo Viale and Christoph Weiß, On the consistency strength of the proper forcing axiom, Adv. Math. 228 (2011), no. 5, 2672–2687. MR 2838054, DOI 10.1016/j.aim.2011.07.016
Christoph Weiß, The combinatorial essence of supercompactness, Ann. Pure Appl. Logic 163 (2012), no. 11, 1710–1717. MR 2959668, DOI 10.1016/j.apal.2011.12.017
Christoph Weiss, Subtle and ineffable tree properties, Ph.D. thesis, Ludwig Maximilians Universität München, 2010.