Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle ${\mathrm{\text{SR}}}^{-}$ introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle ${\mathrm{\text{SR}}}^{-}$ for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak ${\mathrm{\Pi }}_n^1$-indescribability, can be canonically characterized through localized versions of the principle ${\mathrm{\text{SR}}}^{-}$. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak ${\mathrm{\Pi }}_1^1$-indescribability.

受 Bagaria、Magidor 和 Väänänen 的结果的启发，我们通过结构类别的反射原理研究大基数属性的表征。更具体地说，我们的目标是通过原则从大基数层次结构的低端描述概念 ${\mathrm{\text{SR}}}^{-}$ 由 Bagaria 和 Väänänen 介绍。我们的结果在大基数层次结构中分离出一个狭窄的区间，从下到上是完全不可描述的，从上到上是微妙的，并且包含所有可以通过原理的有效性来表征的大基数 ${\mathrm{\text{SR}}}^{-}$ 对于由 Lévy 层次结构中固定级别的公式定义的所有结构类别。此外，事实证明，任何可以通过该原则表征的属性都不能证明是强不可访问性。这些结果的证明在很大程度上依赖于 Rathjen 在证明理论上下文中引入的精明红衣主教的概念，以及嵌入这些红衣主教的表征，类似于 Magidor 对超紧致性的经典表征。此外，我们展示了几个重要的弱大基数属性，如弱不可访问性、弱 Mahloness 或弱 ${\mathrm{\Pi }}_n^1$-不可描述性，可以通过该原理的本地化版本来规范地表征 ${\mathrm{\text{SR}}}^{-}$. 最后，在这些表征的证明中开发的技术也使我们能够证明 Hamkin 的弱紧致嵌入属性等价于 Lévy 的弱概念 ${\mathrm{\Pi }}_1^1$-难以形容。