The tree property at the double successor of a measurable cardinal κ with 2^κlarge

Abstract

Assuming the existence of a λ + -hypermeasurable cardinal κ, where λ is the first weakly compact cardinal above κ, we prove that, in some forcing extension, κ is still measurable, κ ++ has the tree property and 2 κ = κ +++ . If the assumption is strengthened to the existence of a θ-hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ) then the proof can be generalized to get 2 κ = θ.

The Tree Property at And

“…Second, we can produce models for the tree property at κ ++ where κ is singular strong limit and 2 κ > κ ++ . Friedman and Halilovic [4] asked whether there is such a model. Recent work of Friedman, Honzik and Stejskalová [5] gives a positive result.…”