2023/02/06 by Mitch Rudominer, Rudominer, Mitch
Computer Science · Mathematics · #03E15 #03E55 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2302.02581
openalex publication_date 2023/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We identify a particular mouse, Mld, the minimal ladder mouse, that sits in the mouse order just past Mn\sharp for all n, and we show that ℝ∩ Mld = Qω+1, the set of reals that are Δ1ω+1 in a countable ordinal. Thus Qω+1 is a mouse set. This is analogous to the fact that ℝ∩ M\sharp1 = Q3 where M\sharp1 is the the sharp for the minimal inner model with a Woodin cardinal, and Q3 is the set of reals that are Δ13 in a countable ordinal. More generally ℝ∩ M\sharp2n+1 = Q2n+3. The mouse Mld and the set Qω+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∩ Mld ⊆ Qω+1 was known in the 1990's. But Qω+1 ⊆ Mld was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin's proof.