On the limits of determinacy in third-order arithmetic and extensions of Kripke-Platek set theory : an introductive study of topics in reverse mathematics, constructibility and determinacy

This thesis explores the foundations of mathematics, studying determinacy axioms derived from game theory. It exposes their relationship with second-order and third-order arithmetic, examining a significant paper in the field by Montalbán and Shore. The thesis begins by introducing the background concepts of determinacy and second-order arithmetic. It also explains the translation between set theory and second-order arithmetic. Subsequently, the thesis delves into advanced concepts in set theory and subsystems of second-order arithmetic that are crucial for understanding and proving the results proved by Montalbán and Shore. In particular, it presents a generalisation of their second theorem, formulated within a Kripke-Platek set-theoretic version of third-order arithmetic, offering an original contribution to the field. Overall, this thesis contributes to understanding determinacy axioms and their limitations. It provides a synthesis of the foundational knowledge required to comprehend the research conducted by Montalbán and Shore in this specific area.