The mouse set theorem just past projective

We identify a particular mouse, [Formula: see text], the minimal ladder mouse, that sits in the mouse order just past [Formula: see text] for all [Formula: see text], and we show that [Formula: see text], the set of reals that are [Formula: see text] in a countable ordinal. Thus [Formula: see text] is a mouse set. This is analogous to the fact that [Formula: see text] where [Formula: see text] is the sharp for the minimal inner model with a Woodin cardinal, and [Formula: see text] is the set of reals that are [Formula: see text] in a countable ordinal. More generally [Formula: see text]. The mouse [Formula: see text] and the set [Formula: see text] 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. [Formula: see text] was known in the 1990s. But [Formula: see text] was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin’s proof.