Ordinal analysis is the branch of proof theory that aims at quantifying the strength of a mathematical theory or statement by assigning to it a "number." These numbers are ordinal numbers, natural extensions of the counting number to the transfinite. This number, the proof-theoretic ordinal of a mathematical theory, is the least number that one cannot construct from the natural numbers in a recursive way via means provided by the theory. All known computations of proof-theoretic ordinals yield strong insight into the theories in question. For instance, every known computation has been accompanied by a consistency proof--a proof of the fact that the theory leads to no contradictions--starting from the assumption that a re-ordering of the natural numbers of length the proof-theoretic ordinal exists.

Given a class Gamma of infinite two-player, perfect-information games on the natural numbers, the axiom of Gamma-determinacy states that all games in Gamma are determined, i.e., one of the players has a winning strategy. When Gamma is the class of all games, this axiom is inconsistent with the Axiom of Choice; however, when Gamma is some definable collection of games, Gamma-determinacy is not only consistent, but true. However, proofs of the axiom for various Gamma are complicated and thus are expected to have very large proof-theoretic ordinals. This project is concerned with the computation of the proof theoretic ordinal for Gamma-determinacy, for various simple Gamma.