Archive for Mathematical Logic 50 (5-6):585-590 (2011)

Abstract
We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “ ${{\rm ZF} + \neg{\rm AC}_\omega}$ + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from hypotheses stronger in consistency strength than a supercompact limit of supercompact cardinals. A lower bound in consistency strength is provided by a result of Busche and Schindler, who showed that the consistency of the theory “ZF + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” implies the consistency of AD L(R)
Keywords Supercompact cardinal  Tree property  Indestructibility  Symmetric inner model
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DOI 10.1007/s00153-011-0233-z
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References found in this work BETA

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
Gap Forcing: Generalizing the Lévy-Solovay Theorem.Joel David Hamkins - 1999 - Bulletin of Symbolic Logic 5 (2):264-272.
Aronszajn Trees and the Independence of the Transfer Property.William Mitchell - 1972 - Annals of Pure and Applied Logic 5 (1):21.
The Tree Property at Successors of Singular Cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
Aronszajn Trees on ℵ2 and ℵ3.Uri Abraham - 1983 - Annals of Pure and Applied Logic 24 (3):213-230.

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