Thesis Summary

The main goal of my thesis is the application of logical and computability-theoretic techniques to better understand the foundational nature of structures and theorems from different branches of mathematics. To achieve this goal, I examined the effective (i.e. computable) content of structures and theorems from several different branches of math, including computability theory and model theory (Chapter 1), fractal dimension and probability theory (Chapter 2), commutative and noncommutative rings (Chapters 3 and 4), and measure theory (Chapter 5). My thesis contributes to these areas of mathematics by examining the effective content of central structures and/or theorems in each of them, and in so doing sheds light on their fundamental (i.e. logical) nature. The next five sections describe the five chapters of my thesis in sequence.