A remark on the tree property in a choiceless context

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 “ZF + ¬ACω + 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) . We begin by very briefly mentioning some of our conventions and terminology. As usua...

• Type: Journal
• Year: 2011
• Where: AML
• Authors: Arthur W. Apter
• Keywords: AML 2011, Busche, Consistency Strength, Mathematics, Satis