2021/02/18 by Brent Cody, Cody, Brent
Mathematics · #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2102.09598
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 Lκ+,κ+-indescribability and Π1ξ-indescribability of a cardinal κ for all ξ<κ+. In this context, universal Π1ξ formulas exist, there is a normal ideal associated to Π1ξ-indescribability and the notions of Π1ξ-indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given a regular cardinal μ, we introduce a diagonal version of Cantor's derivative operator and use it to extend Bagaria's \citeMR3894041 sequence langleτξ:ξ<μ⟩ of derived topologies on μ to ⟨τξ:ξ<μ+⟩. Finally, we prove that for all ξ<μ+, if there is a stationary set of α<μ that have a high enough degree of indescribability, then there are stationarily-many α<μ that are nonisolated points in the space (μ,τξ+1).