Derived topologies on ordinals and stationary reflection

Bagaria, Joan

doi:10.1090/tran/7366

Derived topologies on ordinals and stationary reflection
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Trans. Amer. Math. Soc. 371 (2019), 1981-2002 Request permission

Abstract:

We study the transfinite sequence of topologies on the ordinal numbers that is obtained through successive closure under Cantor’s derivative operator on sets of ordinals, starting from the usual interval topology. We characterize the non-isolated points in the $\xi$th topology as those ordinals that satisfy a strong iterated form of stationary reflection, which we call $\xi$-simultaneous-reflection. We prove some properties of the ideals of non-$\xi$-simultaneous-stationary sets and identify their tight connection with indescribable cardinals. We then introduce a new natural notion of $\Pi ^1_\xi$-indescribability, for any ordinal $\xi$, which extends to the transfinite the usual notion of $\Pi ^1_n$-indescribability, and prove that in the constructible universe $L$, a regular cardinal is $(\xi +1)$-simultaneously-reflecting if and only if it is $\Pi ^1_\xi$-indescribable, a result that generalizes to all ordinals $\xi$ previous results of Jensen [Ann. Math. Logic 4 (1972), pp. 229–308] in the case $\xi =2$, and Bagaria–Magidor–Sakai [Israel J. Math. 208 (2015), pp. 1–11] in the case $\xi =n$. This yields a complete characterization in $L$ of the non-discreteness of the $\xi$-topologies, both in terms of iterated stationary reflection and in terms of indescribability.

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