The tree property at the aleph_2n's and the failure of SCH at aleph_omega

Abstract

We show - starting from a hypermeasurable-type large cardinal assumption - that one can force a model where 2אω = אω+2, אω is a strong limit cardinal, and the tree property holds at all א2n, for n > 0. This provides a partial answer to the question whether the failure of SCH at אω is consistent with many cardinals below אω having the tree property.

Keywords

• tree property
• hypermeasurable cardinals