Woodin cardinals fall between strong and supercompact. Every supercompact cardinal is Woodin, and if \(\delta\) is Woodin, then \(V_\delta\) is a model of ZFC in which there is a proper class of strong cardinals. Thus, while a Woodin cardinal \(\delta\) need not be itself very strong—the first one is not even weakly compact—it implies the existence of many large cardinals in \(V_\delta\).