A Diamond Principle Consistent with AD

Abstract

We present a diamond principle ${◊}_{\mathbb{R}}$ concerning all subsets of $\Theta$, the supremum of the ordinals that are the surjective image of $\mathbb{R}$. We prove that ${◊}_{\mathbb{R}}$ holds in Steel’s core model $\mathbf{K}\left(\mathbb{R}\right)$, a canonical inner model for determinacy.