Classifying Derived Models of the Axiom of Determinacy

The research questions in this project are at the boundaries of mathematics, more precisely, at the boundaries of what can be proved in mathematics. Inspired by Kurt Gödels famous incompleteness theorems, a central goal of research in set theory is to classify and understand statements that can neither be proven nor disproven in the standard axiom system for mathematics. A well-known statement that in this sense can neither be proven nor disproven is the existence of winning strategies in infinitely long two-player-games. More precisely, we consider games in which two players alternate choosing natural numbers. Which player wins the game depends on the infinite sequence of natural numbers that the two players jointly produced during a run of the game. The game is called determined if one of the players has a winning strategy, that means a strategy that allows them to win every run of the game if they are playing according to the strategy (depended on the moves of their opponent). We study models of set theory in which every such two-player-game is determined. A famous method to construct such models of determinacy was developed by Hugh Woodin in the 1980s based on his proof of the so-called Derived Model Theorem. We study extensions of this method and aim to classify such derived models. Different than before, we will consider derived models which have a complex structure also at very large infinities. Such a model was, for example, recently used by Paul Larson and Grigor Sargsyan to refute the iterability conjecture. This suggests that a better understanding of such models is central for further progress in this area.