THE INEFFABLE TREE PROPERTY and FAILURE of the SINGULAR CARDINALS HYPOTHESIS

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

doi:10.1090/tran/8110

THE INEFFABLE TREE PROPERTY and FAILURE of the SINGULAR CARDINALS HYPOTHESIS

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

Research output: Contribution to journal › Article › peer-review

Abstract

ITP is a combinatorial principle that is a strengthening of the tree property. For an inaccessible cardinal κ, ITP at κ holds if and only if κ 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 κ together with ITP κ+. Then we bring down the result to κ = ? ω2 .

Original language	American English
Pages (from-to)	5937-5955
Number of pages	19
Journal	Transactions of the American Mathematical Society
Volume	373
Issue number	8
DOIs	https://doi.org/10.1090/tran/8110
State	Published - Aug 2020
Externally published	Yes

Bibliographical note

Funding Information:
Received by the editors April 4, 2019, and, in revised form, January 26, 2020. 2010 Mathematics Subject Classification. Primary 03E05, 03E35, 03E55. The first author was partially supported by the National Science Foundation, DMS-1500790. The second author was partially supported by FWF, M 2650 Meitner-Programm. The fourth author was partially supported by the National Science Foundation, DMS-1764029. The fifth author was partially supported by the National Science Foundation, Career-1454945. The sixth author was partially supported by the National Science Foundation, DMS-1700425.

Publisher Copyright:
© 2020 American Mathematical Society. All rights reserved.

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abstract = "ITP is a combinatorial principle that is a strengthening of the tree property. For an inaccessible cardinal κ, ITP at κ holds if and only if κ 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 κ together with ITP κ+. Then we bring down the result to κ = ? ω2 .",

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note = "Funding Information: Received by the editors April 4, 2019, and, in revised form, January 26, 2020. 2010 Mathematics Subject Classification. Primary 03E05, 03E35, 03E55. The first author was partially supported by the National Science Foundation, DMS-1500790. The second author was partially supported by FWF, M 2650 Meitner-Programm. The fourth author was partially supported by the National Science Foundation, DMS-1764029. The fifth author was partially supported by the National Science Foundation, Career-1454945. The sixth author was partially supported by the National Science Foundation, DMS-1700425. Publisher Copyright: {\textcopyright} 2020 American Mathematical Society. All rights reserved.",

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doi = "10.1090/tran/8110",

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AU - Sinapova, Dima

AU - Spencer,

N1 - Funding Information: Received by the editors April 4, 2019, and, in revised form, January 26, 2020. 2010 Mathematics Subject Classification. Primary 03E05, 03E35, 03E55. The first author was partially supported by the National Science Foundation, DMS-1500790. The second author was partially supported by FWF, M 2650 Meitner-Programm. The fourth author was partially supported by the National Science Foundation, DMS-1764029. The fifth author was partially supported by the National Science Foundation, Career-1454945. The sixth author was partially supported by the National Science Foundation, DMS-1700425. Publisher Copyright: © 2020 American Mathematical Society. All rights reserved.

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THE INEFFABLE TREE PROPERTY and FAILURE of the SINGULAR CARDINALS HYPOTHESIS

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