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COUNTABLE STATIONARY TOWERS AND REFLECTION PRINCIPLES
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 15 October 2026, pp. 1-11
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- Article
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In this article, we prove that the precipitousness of $\mathbb {Q}_\gamma $
, the countable stationary tower of height $\gamma $
, where $\gamma $
is a limit ordinal $>\omega $
, is a consequence of a certain stationary reflection principle for $\mathcal {P}_{\omega _1} H_{\left \vert {V_\gamma } \right \vert {}^+}$
. Using this, we prove the consistency of the precipitousness of $\mathbb {Q}_\gamma $
for some singular $\gamma $
assuming a sufficiently strong large cardinal axiom. Furthermore, assuming the existence of proper class many supercompact cardinals, we build a model where $\mathbb {Q}_\gamma $
is precipitous for every limit ordinal $>\omega $
.
