Guessing models and the approachability ideal

Abstract

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call [Formula: see text] holds. This principle implies [Formula: see text] and [Formula: see text], and hence the tree property at [Formula: see text] and [Formula: see text], the Singular Cardinal Hypothesis, and the failure of the weak square principle [Formula: see text], for all regular [Formula: see text]. In addition, it implies that the restriction of the approachability ideal [Formula: see text] to the set of ordinals of cofinality [Formula: see text] is the nonstationary ideal on this set. The consistency of this last statement was previously shown by W. Mitchell.