A remark on the tree property in a choiceless context

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 "ZF + ¬AC ω + 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) .We begin by very briefly mentioning some of our conventions and terminology. As usual, for κ a regular cardinal, a κ-tree is a tree of height κ, all of whose levels have cardinality less than κ. κ satisfies the tree property if every κ-tree has a branch of length κ. A κ-tree not satisfying the tree * 2000 Mathematics Subject Classifications: 03E25, 03E35, 03E45, 03E55.†

Remarks on Gitik's model and symmetric extensions on products of the Lévy collapse

“…Apter constructed a symmetric extension based on Lévy collapses in [5] where every successor cardinal has tree property. Applying [24,Theorem 4.3], we can further claim that "Every successor cardinal has tree property which does not carry any uniform ultrafilter".…”