Tame failures of the unique branch hypothesis and models of AD(R) + Theta is regular

摘要

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame iteration trees implies that in some homogenous generic extension of V there is a transitive model M containing Ord boolean OR R such that M satisfies AD(R) + Theta is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66(4) (2014) 903-923; Core models with more Woodin cardinals, J. Symbolic Logic 67(3) (2002) 1197-1226] for tame trees.