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.

我们识别出一只特定的老鼠，${\mathrm{中号}}^{\mathrm{\text{LD}}}$，最小的梯形鼠标，位于刚刚过去的鼠标顺序中${\mathrm{中号}}_n^{♯}$对全部$n$，我们证明$ℝ\cap {\mathrm{中号}}^{\mathrm{\text{LD}}}={问}_{\omega +1}$，实数集是${\mathrm{\Delta }}_{\omega +1}^1$在可数序数词中。因此${问}_{\omega +1}$是一套鼠标。

这类似于以下事实：$ℝ\cap {\mathrm{中号}}_1^{♯}={问}_3$在哪里${\mathrm{中号}}_1^{♯}$是带有伍丁基数的最小内部模型的尖锐，并且${问}_3$是一组实数${\mathrm{\Delta }}_3^1$在可数序数词中。

更普遍$ℝ\cap {\mathrm{中号}}_{2n+1}^{♯}={问}_{2n+3}$。鼠标${\mathrm{中号}}^{\mathrm{\text{LD}}}$和集合${问}_{\omega +1}$组成下一个自然对以在这一系列结果中考虑。因此，我们证明了刚刚射影的鼠标集定理。

其中一些并不新鲜。$ℝ\cap {\mathrm{中号}}^{\mathrm{\text{LD}}}\subseteq {问}_{\omega +1}$于 20 世纪 90 年代为人所知。但${问}_{\omega +1}\subseteq {\mathrm{中号}}^{\mathrm{\text{LD}}}$直到伍丁在 2018 年找到证明之前一直开放。本文的主要目标是给出伍丁的证明。