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]. (shrink)

For n<ω, we say that theΠn1-reflection principle holds at κ and write Refln if and only if κ is a Πn1-indescribable cardinal and every Πn1-indescribable subset of κ has a Πn1-indescribable proper initial segment. The Πn1-reflection principle Refln generalizes a certain stationary reflection principle and implies that κ is Πn1-indescribable of order ω. We define a forcing which shows that the converse of this implication can be false in the case n=1; that is, we show that κ being Π11-indescribable of (...) order ω need not imply Refl1. Moreover, we prove that if κ is -weakly compact where α<κ+, then there is a forcing extension in which there is a weakly compact set W⊆κ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and κ remains -weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down. (shrink)