The tree property at the double successor of a measurable cardinal $\kappa $ with $2^{\kappa} $ large
Volume 223 / 2013
Fundamenta Mathematicae 223 (2013), 55-64
MSC: 03E35, 03E55.
DOI: 10.4064/fm223-1-4
Abstract
Assuming the existence of a $\lambda ^+$-hypermeasurable cardinal $\kappa $, where $\lambda $ is the first weakly compact cardinal above $\kappa $, we prove that, in some forcing extension, $\kappa $ is still measurable, $\kappa ^{++}$ has the tree property and $2^\kappa =\kappa ^{+++}$. If the assumption is strengthened to the existence of a $\theta $-hypermeasurable cardinal (for an arbitrary cardinal $\theta >\lambda $ of cofinality greater than $\kappa $) then the proof can be generalized to get $2^\kappa =\theta $.
