Models of the axiom of determinacy and the foundations of set theory

Abstract

David Hilbert declared that “No one shall expel us from the paradise which Cantor has created for us.” For Cantor’s paradise—set theory—was a rich source of powerful new mathematical ideas about the infinite. But Cantor’s paradise is not a paradise of understanding. Basic questions about infinity cannot be answered using the axioms of set theory even when supplemented with strong ax- ioms of infinity. The main problem in the foundations of set theory is to identify the true principles which answer these questions. There are infinitely many incompatible ways to extend the axioms of set theory, yet only a very small finite number of theories are pursued as foundational frameworks, possible solutions to the main foundational problem. These include determinacy theories, forcing axioms, canonical inner models, and the Ultimate L framework. While Cantor’s paradise is not a paradise of understanding, models of the Axiom of Determinacy are. They are canonical, and they are neutral: All strong theories, no matter what they mutually disagree on, imply models of determinacy exist. Thus models of determinacy tell us some of what minimally must be the case, no matter which foundational framework we’re working in. This thesis is about (1) applications of models of determinacy to foundational problems in set theory, both philosophical and mathematical, and (2) the mathematical problem of how to obtain larger paradises of understanding. Specifically, we map out a system of conjectures about models of determinacy that, we show, would allow us to address the following foundational questions: 1. Is there a canonical model of the forcing axiom Martin’s Maximum? 2. Is Martin’s Maximum consistent with the principle (∗)++ UB? 3. Can Martin’s Maximum be forced over a model of the Axiom of Determinacy? 4. What kinds of justifications could secure the truth of Martin’s Maximum? 5. Why are large cardinals more powerful in the context of ZFC+V=Ultimate L than in just ZFC? 6. How can the existence of large cardinals above the least supercompact cardinal be justified in light of Woodin’s Universality Theorem? 7. Does V=Ultimate L embody a more general, natural determinacy principle? 8. Can the Axiom of Determinacy be generalized to ℘(℘(R)),℘(℘(℘(R))),...?