A covering lemma for HOD of \(K(\mathbb R)\) (Q609760)

This is the third in the series of the author's articles on covering lemmas for inner models. In the first of the series [Arch. Math. Logic 41, No.~1, 49--54 (2002; Zbl 1022.03031)], the author developed the covering lemmas for \(L({\mathbb R})\), as well as \(\text{HOD}^{L({\mathbb R})}\), under the assumption of \(\text{ZF}+\text{AD}\) and the nonexistence of \({\mathbb R}^\sharp\). In the second [Arch. Math. Logic 46, No. 3--4, 197--221 (2007; Zbl 1110.03044)] and this paper, the author generalizes these results to an inner model called \(K({\mathbb R})\). \(K({\mathbb R})\) is built with mice that are associated to \({\mathbb R}\)-complete measures, and is less general than Steel's \(\mathbf{K}({\mathbb R})\), which is built with mice that are associated to sequence of extenders. This paper proves the covering lemma for \(\text{HOD}^{K({\mathbb R})}\). The main theorem states that under \(\text{ZF}+\text{AD}\) and the nonexistence of an inner model with \({\mathbb R}\)-complete measurable cardinals, every set of reals of size \(\geq \Theta\) (the supremum of the ordinals that are the surjective images of \({\mathbb R}\)) can be covered by one in \(\text{HOD}^{K({\mathbb R})}\) of the same size. A key ingredient of the argument is the version of a theorem of Woodin's adapted to \(K({\mathbb R})\), more precisely, \(K({\mathbb R})\) is a symmetric generic extension of its own \(\text{HOD}\). Though this follows from Steel's analysis for \(\mathbf{K}({\mathbb R})\), the author provides his own arguments. As a corollary, the author obtains that, assuming \(\text{ZF}+\text{AD}\), the existence of an inner model with \({\mathbb R}\)-complete measurable cardinals is equivalent to the existence of a set of ordinals that has size \(\geq \Theta^{K({\mathbb R})}\) but no covering set in \(\text{HOD}^{K({\mathbb R})}\).