An undecidable extension of Morley's theorem on the number of countable models

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of $\sigma$-projective equivalence relations in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals and Inner Model Theory.Comment: 31 page

The Mouse Set Theorem Just Past Projective

We identify a particular mouse, $M^{\text{ld}}$, the minimal ladder mouse, that sits in the mouse order just past $M_n^{\sharp}$ for all $n$, and we show that $\mathbb{R}\cap M^{\text{ld}} = Q_{\omega+1}$, the set of reals that are $\Delta^1_{\omega+1}$ in a countable ordinal. Thus $Q_{\omega+1}$ is a mouse set. This is analogous to the fact that $\mathbb{R}\cap M^{\sharp}_1 = Q_3$ where $M^{\sharp}_1$ is the the sharp for the minimal inner model with a Woodin cardinal, and $Q_3$ is the set of reals that are $\Delta^1_3$ in a countable ordinal. More generally $\mathbb{R}\cap M^{\sharp}_{2n+1} = Q_{2n+3}$. The mouse $M^{\text{ld}}$ 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. $\mathbb{R}\cap M^{\text{ld}} \subseteq Q_{\omega+1}$ was known in the 1990's. But $Q_{\omega+1} \subseteq M^{\text{ld}}$ was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin's proof.Comment: 30 page