Reflection Spectra: Predicative Mathematics and Beyond

By work of Austrian logician Kurt Gödel in the 1930s, no mathematical theory which is sufficiently strong and correct can prove its own consistency. Soon after Gödels work, G. Gentzen provided a proof of the consistency of arithmetic that employed one sole extraneous component: a use of transfinite induction up to a suitable infinite number. Ordinal analysis is the branch of Proof Theory that studies generalisations of Gentzens theorem whereby one identifies larger and larger infinite numbers and extracts from them crucial information about a mathematical theory. Given a mathematical theory, one can find a family of infinite numbers that capture various parameters of information about the theory this is the reflection spectrum of the theory. The goal of this project is to identify and study the reflection spectra of various mathematical theories of increasing strength. Along the way, we expect to isolate various arithmetical principles and logics of independent interest.