# Is there a set of reals not inK(R)?

- Authors
- Journal
- Annals of Pure and Applied Logic 0168-0072
- Publisher
- Elsevier
- Publication Date
- Volume
- 92
- Issue
- 2
- Identifiers
- DOI: 10.1016/s0168-0072(98)00003-7
- Keywords

## Abstract

Abstract 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 HOD K ( 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 + ¬ DC R implies that R † (dagger) exists.

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