Fine Hierarchies and Laver Tables: Complexity in Arithmetic and Set Theory

This thesis investigates two major topics: the extension of Selivanov’s fine hierarchy to hyperarithmetic sets (and the associated Turing degrees), and the lower bound estimates for critical points in Laver tables arising from large cardinal axioms. In the first part, we show how to extend Selivanov’s fine hierarchy using descriptions of Borel Wadge classes. We give a game characterisation of containment between classes. We show that every class in the extended fine hierarchy has an admissible description, and use this to calculate heights in the hierarchy. We also answer the question: which levels of Selivanov’s fine hierarchy contain new Turing degrees? In the second part, we present some new lower bound estimates for certain numbers in Laver table theory and introduce several related structures of interest.