Reflecting measures

Abstract

We give new, purely combinatorial characterizations of several kinds of large cardinals, such as strongly $C^{(n)}$-compact and $C^{(n)}$-extendible, in terms of reflecting measures. We then study the key property of tightness of elementary embeddings that witness strong $C^{(n)}$-compactness, which prompts the introduction of the new large-cardinal notion of tightly $C^{(n)}$-compact cardinal. Then we prove, assuming the Ultrapower Axiom, that a cardinal is tightly $C^{(n)}$-compact if and only if it is either $C^{(n-1)}$-extendible or a measurable limit of $C^{(n-1)}$-extendible cardinals. In the last section we also give new characterizations of $\Sigma_n$-strong cardinals in terms of reflecting extenders.