WOODIN FOR STRONG COMPACTNESS CARDINALS

Woodin and Vopěnka cardinals are established notions in the large cardinal hierarchy and it is known that Vopěnka cardinals are the Woodin analogue for supercompactness. Here we give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor’s identity crisis theorem for the first strongly compact cardinal.