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The Main Theorem is the equiconsistency of the following two statements: (1) is a measurable cardinal and the tree property holds at ++; (2) is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the double successor of each strongly inaccessible cardinal. If 0# exists, then we can construct an inner model in which the tree property holds at the double successor of each strongly inacces- sible cardinal. We also nd upper and lower bounds for the consistency strength of there being no special Aronszajn trees at the double succes- sor of a measurable cardinal. The upper and lower bounds dier only by 1 in the Mitchell order.

The consistency strength of the tree property at the double successor of a measurable cardina