Abstract

Journal of Mathematical Logic, Ahead of Print. We identify a particular mouse, [math], the minimal ladder mouse, that sits in the mouse order just past [math] for all [math], and we show that [math], the set of reals that are [math] in a countable ordinal. Thus [math] is a mouse set. This is analogous to the fact that [math] where [math] is the sharp for the minimal inner model with a Woodin cardinal, and [math] is the set of reals that are [math] in a countable ordinal. More generally [math]. The mouse [math] and the set [math] compose the next natural pair to consider in this series of results. Thus we are proving the mouse set theorem just past projective. Some of this is not new. [math] was known in the 1990s. But [math] was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin’s proof.