In this dissertation we prove results relating to Martin's Conjecture, a foundational conjecture in Recursion Theory, as well as results in Inner Model Theory, a part of Set Theory which plays a central role in the meta-mathematics of Set Theory. While the different parts of the thesis are not closely related, our interest in the work presented here stems from an interest in the foundations of mathematics.

In chapter 1, we prove results relating to Martin's Conjecture which are joint work with Patrick Lutz. Among other things, we show that part of the conjecture holds for a natural class of functions, the order-preserving functions. In chapter 2, we prove uniqueness theorems about the core model, a fundamental object of study in Inner Model Theory. Our theorems identify the core model in elementary set-theoretic terms, whereas the usual definitions of the core model require deep knowledge of Inner Model Theory. Finally, in chapter 3, we develop the theory meta-iteration trees, a framework for studying the kind of iteration tree combinatorics which has become central to the study of mouse pairs and is relevant for applications of Inner Model Theory to Descriptive Set Theory.