Martin’s Maximum and tower forcing

Abstract

There are several examples in the literature showing that compactness-like properties of a cardinal κ cause poor behavior of some generic ultrapowers which have critical point κ (Burke [1] when κ is a supercompact cardinal; Foreman-Magidor [6] when κ = ω 2 in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $$\overrightarrow I $$ is a tower of ideals which concentrates on the class $$GI{C_{{\omega _1}}}$$ of ω 1-guessing, internally club sets, then $$\overrightarrow I $$ is not presaturated (a set is ω 1-guessing iff its transitive collapse has the ω 1-approximation property as defined in Hamkins [10]). This theorem, combined with work from [16], shows that if PFA + or MM holds and there is an inaccessible cardinal, then there is a tower with critical point ω 2 which is not presaturated; moreover, this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor [6]) to exist in all models of Martin’s Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at ω 2 has similar implications for towers of ideals which concentrate on the wider class $$GI{C_{{\omega _1}}}$$ of ω 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 ω 2) is consistent with a precipitous tower on $$GI{C_{{\omega _1}}}$$ .