Countable Stationary Towers and Reflection Principles

In this article, we prove that the precipitousness of Q γ Q_γ double struck upper Q Subscript gamma , the countable stationary tower of height γ γ gamma , where γ γ gamma is a limit ordinal > ω >ω greater than omega , is a consequence of a certain stationary reflection principle for P ω 1 H | V γ | + Pω ₁ H_ V_γ ⁺ script upper P Subscript omega 1 Baseline upper H Subscript StartAbsoluteValue upper V Sub Subscript gamma Subscript EndAbsoluteValue Sub Superscript plus . Using this, we prove the consistency of the precipitousness of Q γ Q_γ double struck upper Q Subscript gamma for some singular γ γ gamma assuming a sufficiently strong large cardinal axiom. Furthermore, assuming the existence of proper class many supercompact cardinals, we build a model where