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 [Formula: see text]-indescribability and [Formula: see text]-indescribability of a cardinal [Formula: see text] for all [Formula: see text]. In this context, universal [Formula: see text] formulas exist, there is a normal ideal associated to [Formula: see text]-indescribability and the notions of [Formula: see text]-indescribability yield a strict hierarchy below a subtle cardinal. Additionally, given a regular cardinal [Formula: see text], we introduce a diagonal version of Cantor’s derivative operator and use it to extend Bagaria’s [Derived topologies on ordinals and stationary reflection, Trans. Amer. Math. Soc. 371(3) (2019) 1981–2002] sequence [Formula: see text] of derived topologies on [Formula: see text] to [Formula: see text]. Finally, we prove that for all [Formula: see text], if there is a stationary set of [Formula: see text] that have a high enough degree of indescribability, then there are stationarily many [Formula: see text] that are nonisolated points in the space [Formula: see text].