The Metamathematics Of Separated Determinacy J P Aguilera by J. P. Aguilera 101007/S00222025013223 instant download after payment.

Abstract

Determinacy axioms are mathematical principles which assert that various infinitegames are determined. In this article, we prove three general meta-theorems on thelogical strength of determinacy axioms. These allow us to reduce a metamathematicalanalysis of the principle of Γ-determinacy over a weak base theory to an analysis of aprinciple Γ′-determinacy, where Γ′ is a strictly smaller complexity class than Γ. Themeta-theorems are proved in the weak theory RCA0. However, they are formulatedin 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 optimalstrengthenings of the transfer theorems of Martin-Harrington, Kechris-Woodin, andNeeman. As the main application of the meta-theorems, we carry out all the reversemathematical analyses of theories of determinacy below T = 11−CA0 + Π14−CA0which are missing from the literature. More precisely, let Γ ⊂ P(R) be called aWadge class if Γ is closed under continuous preimages. For each Wadge class Γsuch that the consistency of Γ-Determinacy is provable in T , we reduce the principleof Γ-Determinacy to a combination of Comprehension, Monotone Induction, and βReflection axioms. It follows from our results that these classes Γ are precisely thosewhich satisfyo(Γ) < ωω211 and T ⊢“Γ is a Wadge class.”Our work extends and generalizes results of Friedman, Hachtman, Heinatsch, Martin,MedSalem, Montalbán, Möllerfeld, Nemoto, Shore, Steel, Tanaka, Welch, and others,and concludes the project of metamathematical analysis of determinacy principles(cf. e.g., Montalbán’s “Open Questions in Reverse Mathematics”, Bull. Symb. Log.16:431–454, 2011), as far as subsystems of T are concerned.