Ladder mice

Abstract

Assume ZF + AD + $V=L(\mathbb{R})$. We prove some "mouse set" theorems, for definability over $J_\alpha(\mathbb{R})$ where $[\alpha,\alpha]$ is a projective-like gap (of $L(\mathbb{R})$) and $\alpha$ is either a successor ordinal or has countable cofinality, but $\alpha\neq\beta+1$ where $\beta$ ends a strong gap. For such ordinals $\alpha$ and integers $n\geq 1$, we show that there is a mouse $M$ with $\mathbb{R}\cap M=\mathrm{OD}_{\alpha n}$. The proof involves an analysis of ladder mice and their generalizations to $J_\alpha(\mathbb{R})$. This analysis is related to earlier work of Rudominer, Woodin and Steel on ladder mice. However, it also yields a new proof of the mouse set theorem even at the least point where ladder mice arise -- one which avoids the stationary tower. The analysis also yields a corresponding "anti-correctness" result on a cone, generalizing facts familiar in the projective hierarchy; for example, that $(\Pi^1_3)^V\upharpoonright M_1$ truth is $(\Sigma^1_3)^{M_1}$-definable and $(\Sigma^1_3)^{M_1}$ truth is $(\Pi^1_3)^V\upharpoonright M_1$-definable. We also define and study versions of ladder mice on a cone at the end of weak gap, and at the successor of the end of a strong gap, and an anti-correctness result on a cone there.