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 ADL(R).
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Arthur W. Apter’s research was partially supported by PSC-CUNY grants. He wishes to thank Ralf Schindler for helpful correspondence on the subject matter of this paper which considerably improved and clarified its presentation.
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Apter, A.W. A remark on the tree property in a choiceless context. Arch. Math. Logic 50, 585–590 (2011). https://doi.org/10.1007/s00153-011-0233-z
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DOI: https://doi.org/10.1007/s00153-011-0233-z