更高的可扩展性和衍生拓扑
Higher indescribability and derived topologies
摘要：
我们介绍了反射性质的基数，其中反映的属性是由无限公式，其长度可以严格大于考虑中的基数表示。这种广义反射原理导致了基数$κ$对所有$ξ<κ^+$的$L_{κ^+,κ^+}$ -可分解性和$Π^1_ξ$ -可分解性的定义。在此上下文中，存在通用的$Π^1_ξ$公式，存在与$Π^1_ξ$ -可度量性相关联的正常理想，并且$Π^1_ξ$ -可度量性的概念在可度量基数以下产生严格的层次结构。另外，给定正则基数$μ$，我们引入了Cantor导算子的对角形式，并利用它将Bagaria的\cite{MR3894041}序列$langleτ_ξ:ξ<μ\rangle$推广到$\langleτ_ξ:ξ<μ^+\rangle$ .最后，证明了对于所有的$ξ<μ^+$，如果存在一个$α<μ$的平稳集具有足够高的可解度，则存在平稳的多个$α<μ$是空间$(μ,τ_{ξ+1})$中的非孤立点.
英文摘要：
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 $L_{κ^+,κ^+}$-indescribability and $Π^1_ξ$-indescribability of a cardinal $κ$ for all $ξ<κ^+$. In this context, universal $Π^1_ξ$ formulas exist, there is a normal ideal associated to $Π^1_ξ$-indescribability and the notions of $Π^1_ξ$-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 \cite{MR3894041} sequence $langleτ_ξ:ξ<μ\rangle$ of derived topologies on $μ$ to $\langleτ_ξ:ξ<μ^+\rangle$. 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 $(μ,τ_{ξ+1})$.
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