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We introduce reflection properties of cardinals in which the attributes thatreflect are expressible by infinitary formulas whose lengths can be strictlylarger than the cardinal under consideration. This kind of generalizedreflection principle leads to the definitions of$L_{\kappa^+,\kappa^+}$-indescribability and $\Pi^1_\xi$-indescribability of acardinal $\kappa$ 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 $\Pi^1_\xi$-indescribabilityyield a strict hierarchy below a measurable cardinal. Additionally, given aregular cardinal $\mu$, we introduce a diagonal version of Cantor's derivativeoperator 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 ofindescribability, then there are stationarily-many $\alpha<\mu$ that arenonisolated points in the space $(\mu,\tau_{\xi+1})$.

Higher indescribability and derived topologies