Martin's Maximum and tower forcing

Sean Cox, Matteo Viale

Published 2011-10-07, updated 2011-10-18Version 3

There are several examples in the literature showing that compactness-like properties of a cardinal $\kappa$ cause poor behavior of some generic ultrapowers which have critical point $\kappa$ (Burke \cite{MR1472122} when $\kappa$ is a supercompact cardinal; Foreman-Magidor \cite{MR1359154} when $\kappa = \omega_2$ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\vec{\mathcal{I}}$ is a tower of ideals which concentrates on the class $GIC_{\omega_1}$ of $\omega_1$-guessing, internally club sets, then $\vec{\mathcal{I}}$ is not presaturated (a set is $\omega_1$-guessing iff its transitive collapse has the $\omega_1$-approximation property as defined in Hamkins \cite{MR2540935}). This theorem, combined with work from \cite{VW_ISP}, shows that if $PFA^+$ or $MM$ holds and there is an inaccessible cardinal, then there is a tower with critical point $\omega_2$ which is not presaturated; moreover this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor \cite{MR1359154}) to exist in all models of Martin's Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at $\omega_2$ has similar implications for towers of ideals which concentrate on the wider class $GIS_{\omega_1}$ of $\omega_1$-guessing, internally stationary sets. Finally, we show that the word "presaturated" cannot be replaced by "precipitous" in the theorems above: Martin's Maximum (which implies SRP and the Tree Property at $\omega_2$) is consistent with a precipitous tower on $GIC_{\omega_1}$.