Is there a set of reals not in K(R)? Original Research Article

We show, using the fine structure of K(R), that the theory ZF + AD + ∃X ⊆ R[X ∉ K(R)] implies the existence of an inner model of ZF + AD + DC containing a measurable cardinal above its Θ, the supremum of the ordinals which are the surjective image of R. As a corollary, we show that HODK(R) = K(P) for some P ⊆ (Θ+)K(R) where K(P) is the Dodd-Jensen Core Model relative to P. In conclusion, we show that the theory ZF + AD + ¬DCR implies that R† (dagger) exists.