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 $L_{{\kappa }^{+},{\kappa }^{+}}$-indescribability and ${\mathrm{\Pi }}_{\xi }^1$-indescribability of a cardinal $\kappa$ for all $\xi <{\kappa }^{+}$. In this context, universal ${\mathrm{\Pi }}_{\xi }^1$ formulas exist, there is a normal ideal associated to ${\mathrm{\Pi }}_{\xi }^1$-indescribability and the notions of ${\mathrm{\Pi }}_{\xi }^1$-indescribability yield a strict hierarchy below a subtle cardinal. Additionally, given a regular cardinal $\mu$, 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 $〈{\tau }_{\xi }:\xi <\mu 〉$ of derived topologies on $\mu$ to $〈{\tau }_{\xi }:\xi <{\mu }^{+}〉$. Finally, we prove that for all $\xi <{\mu }^{+}$, if there is a stationary set of $\alpha <\mu$ that have a high enough degree of indescribability, then there are stationarily many $\alpha <\mu$ that are nonisolated points in the space $(\mu ,{\tau }_{\xi +1})$.

我们引入基数的反射属性，其中反射的属性可以通过无限公式来表达，无限公式的长度可以严格大于所考虑的基数。这种广义反射原理引出了以下定义$L_{{\kappa }^{+},{\kappa }^{+}}$- 不可描述性和${\mathrm{\Pi }}_{\Psi }^1$- 红衣主教的难以形容$\kappa$对全部$\Psi <{\kappa }^{+}$。在此背景下，普遍${\mathrm{\Pi }}_{\Psi }^1$公式存在，有一个正常的理想关联${\mathrm{\Pi }}_{\Psi }^1$- 不可描述性和概念${\mathrm{\Pi }}_{\Psi }^1$- 不可描述性产生了一个严格的等级制度，低于一个微妙的基数。此外，给定一个普通的枢机主教$\mu$，我们引入了康托导数算子的对角版本，并用它来扩展 Bagaria 的[序数和平稳反射的导出拓扑，Trans。阿米尔。数学。苏克。 371（3）（2019）1981-2002]序列$〈{\tau }_{\Psi }：\Psi <\mu 〉$的派生拓扑$\mu$到$〈{\tau }_{\Psi }：\Psi <{\mu }^{+}〉$。最后，我们证明对于所有$\Psi <{\mu }^{+}$，如果有一组平稳的$\alpha <\mu$具有足够高的不可描述性，那么有很多静止的$\alpha <\mu$是空间中的非孤立点$（\mu ,{\tau }_{\Psi +1}）$。