Exact completions; Topos theory; Realizability

We introduce 0•| (“zero hand grenade”) as a sharp for an inner model with a proper class of strong cardinals. We prove the existence of the core model K in the theory “View the MathML source does not exist”. Combined with work of Woodin, Steel, and earlier work of the author, this provides the last step for determining the exact consistency strength of the assumption in the statement of the 12th Delfino problem (cf. (Cabal seminar, 81–85, Springer Lecture Notes in Mathematics, Vol. 1333 (1988) pp. 221–224)).