1996/09/16 by Mitch Rudominer, Rudominer, Mitch
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #math.LO
paper · pdf · doi:10.48550/arxiv.math/9609205
arxiv created 1996/09/16 · openalex publication_date 1996/09/16 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let kappa be the least ordinal alpha such that Lalpha(R) is admissible. Let A be the set of reals x such that x is ordinal definable in Lα(R), for some alpha<kappa. It is well known that (assuming determinacy) A is the largest countable inductive set of reals. Let T be the following theory: ZFC - Replacement + "There exists ω Woodin cardinals which are cofinal in the ordinals." T has consistency strength weaker than that of the theory ZFC + "There exists omega Woodin cardinals", but stronger than that of the theory ZFC + "There exists n Woodin Cardinals", for each n. Let M be the canonical, minimal inner model for the theory T. In this paper we show that A is equal to the set of reals in M. Since M is a "mouse", we say that A is a "mouse set." As an application, we use our characterization of A to give an inner-model-theoretic proof of Martin's theorem that A is equal to the set of reals which are Sigma^*n for some n.