Guessing models and the approachability ideal

TL;DR

The paper constructs a model starting from two supercompact cardinals where a new principle implies several combinatorial and set-theoretic properties, including the tree property, SCH, and specific ideal characteristics.

Contribution

It introduces the principle ${ m GM}^+(oldsymbol{ extomega_3, extomega_1})$ and demonstrates its consistency with various significant set-theoretic properties.

Findings

${ m GM}^+(oldsymbol{ extomega_3, extomega_1})$ holds in the constructed model

The principle implies ${ m ISP}(oldsymbol{ extomega_2})$ and ${ m ISP}(oldsymbol{ extomega_3})$

The approachability ideal $I[oldsymbol{ extomega_2}]$ has a specific non-stationary restriction

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.