The tree property at the double successor of a measurable cardinal $\kappa $ with $2^{\kappa} $ large

Volume 223 / 2013

Sy-David Friedman, Ajdin Halilović Fundamenta Mathematicae 223 (2013), 55-64 MSC: 03E35, 03E55. DOI: 10.4064/fm223-1-4


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 $.


  • Sy-David FriedmanKurt Gödel Research Center
    University of Vienna
    1090 Wien, Austria
  • Ajdin HalilovićFaculty of Engineering Sciences
    Lumina–The University of South East Europe
    021187 Bucureşti, Romania

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