Patterns of Structural Reflection in the large-cardinal hierarchy

Abstract: We unveil new patterns of Structural Reflection in the large-cardinal hierarchy below the first measurable cardinal. Namely, we give two different characterizations of strongly unfoldable and subtle cardinals in terms of a weak form of the principle of Structural Reflection, and also in terms of weak product structural reflection. Our analysis prompts the introduction of the new notion of $C^{(n)}$-strongly unfoldable cardinal for every natural number $n$, and we show that these cardinals form a natural hierarchy between strong unfoldable and subtle cardinals analogous to the known hierarchies of $C^{(n)}$-extendible and $\Sigma_n$-strong cardinals. These results show that the relatively low region of the large-cardinal hierarchy comprised between the first strongly unfoldable and the first subtle cardinals is completely analogous to the much higher region between the first strong and the first Woodin cardinals, and also to the much further upper region of the hierarchy ranging between the first supercompact and the first Vop\v{e}nka cardinals.