Higher indescribability and derived topologies

Abstract

We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of $L_{\kappa^+,\kappa^+}$-indescribability and $\Pi^1_\xi$-indescribability of a cardinal $\kappa$ for all $\xi<\kappa^+$. In this context, universal $\Pi^1_\xi$ formulas exist, there is a normal ideal associated to $\Pi^1_\xi$-indescribability and the notions of $\Pi^1_\xi$-indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given a regular cardinal $\mu$, we introduce a diagonal version of Cantor's derivative operator and use it to extend Bagaria's \cite{MR3894041} sequence $langle\tau_\xi:\xi<\mu\rangle$ of derived topologies on $\mu$ to $\langle\tau_\xi:\xi<\mu^+\rangle$. Finally, we prove that for all $\xi<\mu^+$, if there is a stationary set of $\alpha<\mu$ that have a high enough degree of indescribability, then there are stationarily-many $\alpha<\mu$ that are nonisolated points in the space $(\mu,\tau_{\xi+1})$.