Structural reflection, shrewd cardinals and the size of the continuum

Abstract

Motivated by results of Bagaria, Magidor and V\"a\"an\"anen, 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 $\mathrm{SR}^-$ introduced by Bagaria and V\"a\"an\"anen. 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 $\mathrm{SR}^-$ for all classes of structures defined by formulas in a fixed level of the L\'{e}vy 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 $\Pi^1_n$-indescribability, can be canonically characterized through localized versions of the principle $\mathrm{SR}^-$. 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 L\'{e}vy's notion of weak $\Pi^1_1$-indescribability.