The mouse set theorem just past projective

Abstract

We identify a particular mouse, $M^{\mathrm{\text{ld}}}$, the minimal ladder mouse, that sits in the mouse order just past $M_n^{♯}$ for all $n$, and we show that $ℝ\cap M^{\mathrm{\text{ld}}}=Q_{\omega +1}$, the set of reals that are ${\mathrm{\Delta }}_{\omega +1}^1$ in a countable ordinal. Thus $Q_{\omega +1}$ is a mouse set.

This is analogous to the fact that $ℝ\cap M_1^{♯}=Q_3$ where $M_1^{♯}$ is the sharp for the minimal inner model with a Woodin cardinal, and $Q_3$ is the set of reals that are ${\mathrm{\Delta }}_3^1$ in a countable ordinal.

More generally $ℝ\cap M_{2n+1}^{♯}=Q_{2n+3}$. The mouse $M^{\mathrm{\text{ld}}}$ and the set $Q_{\omega +1}$ 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. $ℝ\cap M^{\mathrm{\text{ld}}}\subseteq Q_{\omega +1}$ was known in the 1990s. But $Q_{\omega +1}\subseteq M^{\mathrm{\text{ld}}}$ was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin’s proof.