COUNTABLE STATIONARY TOWERS AND REFLECTION PRINCIPLES

Abstract In this article, we prove that the precipitousness of Q γ $\mathbb {Q}_\gamma $ double struck upper Q Subscript gamma , the countable stationary tower of height γ $\gamma $ gamma , where γ $\gamma $ gamma is a limit ordinal > ω $>\omega $ greater than omega , is a consequence of a certain stationary reflection principle for P ω 1 H | V γ | + $\mathcal {P}_{\omega _1} H_{\left \vert {V_\gamma } \right \vert {}^+}$ 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 $\mathbb {Q}_\gamma $ double struck upper Q Subscript gamma for some singular γ $\gamma $ gamma assuming a sufficiently strong large cardinal axiom. Furthermore, assuming the existence of proper class many supercompact cardinals, we build a model where