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 ${\rm GM}^+(\omega_3,\omega_1)$ holds. This principle implies ${\rm ISP}(\omega_2)$ and ${\rm ISP}(\omega_3)$, and hence the tree property at $\omega_2$ and $\omega_3$, the Singular Cardinal Hypothesis, and the failure of the weak square principle $\square(\omega_2,\lambda)$, for all regular $\lambda \geq \omega_2$. In addition, it implies that the restriction of the approachability ideal $I[\omega_2]$ to the set of ordinals of cofinality $\omega_1$ is the non stationary ideal on this set. The consistency of this last statement was previously shown by Mitchell.