The metamathematics of separated determinacy

• 成果类型: Article
• 署名作者: Aguilera, J. P.
• 署名单位: Technische Universitat Wien; University of Vienna; Ghent University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01322-3
• 发表日期: 2025
• 页码: 313-457
• 关键词: ramseys theorem; weak axioms; subsystems; strength; EQUIVALENCE; Consistency; cardinals; LIMITS; PROOF; ORDER

摘要

Determinacy axioms are mathematical principles which assert that various infinite games are determined. In this article, we prove three general meta-theorems on the logical strength of determinacy axioms. These allow us to reduce a metamathematical analysis of the principle of Gamma-determinacy over a weak base theory to an analysis of aprinciple Gamma '-determinacy, where Gamma ' is a strictly smaller complexity class than Gamma.Themeta-theorems are proved in the weak theoryRCA0. However, they are formulated in a general way and also have applications in the context of ZFC, eventually leading to an optimal generalization of Martin's Borel determinacy theorem and optimal strengthenings of the transfer theorems of Martin-Harrington, Kechris-Woodin, and Neeman. As the main application of the meta-theorems, we carry out all the reverse-mathematical analyses of theories of determinacy below T=Pi(1)(1)-CA(0)+Pi(1)(4)-CA(0)which are missing from the literature. More precisely, let Gamma subset of P(R)be called aWadge classif Gamma is closed under continuous preimages. For each Wadge class Gamma such that the consistency of Gamma-Determinacy is provable in T, we reduce the principleof Gamma-Determinacy to a combination of Comprehension, Monotone Induction, and beta-Reflection axioms. It follows from our results that these classes Gamma are precisely those which satisfy o(Gamma) < omega(omega 21)(1)and T proves Gamma is a Wadge class. Our work extends and generalizes results of Friedman, Hachtman, Heinatsch, Martin, MedSalem, Montalb & aacute;n, M & ouml;llerfeld, Nemoto, Shore, Steel, Tanaka, Welch, and others, and concludes the project of metamathematical analysis of determinacy principles ,