Ladder mice

Abstract

Assume ZF + AD + V=L(ℝ). We prove some "mouse set" theorems, for definability over J_α(ℝ) where [α,α] is a projective-like gap (of L(ℝ)) and α is either a successor ordinal or has countable cofinality, but α ≠ β+1 where β ends a strong gap. For such ordinals α and integers n ≥ 1, we show that there is a mouse M with ℝ∩M=OD_{αn}. The proof involves an analysis of ladder mice and their generalizations to J_α(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 (Π^1_3)^V↾M_1 truth is (Σ^1_3)^{M_1}-definable and (Σ^1_3)^{M_1} truth is (Π^1_3)^V↾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.