Operations on a Wadge-Type Hierarchy of Ordinal-Valued Functions
Alexander C. Block
Abstract:
In this thesis we investigate – under the assumption of the Axiom of Determinacy (AD) – the structure of the hierarchy of regular norms, a Wadge-type hierarchy of ordinal-valued functions that originally arose from Moschovakis’s proof of the First Periodicity Theorem in descriptive set theory. We introduce the notion of a very strong better quasi-order that will provide a framework to treat the hierarchy of regular norms and the well-known Wadge hierarchy uniformly. From this we can establish classical results for both hierarchies with uniform proofs. Among these are the Martin-Monk and the Steel-Van Wesep Theorem. After that we define operations on the hierarchy of regular norms which are used to show closure properties of this hierarchy. Using these closure properties, we can significantly improve the best formerly known lower bound for the order type of the hierarchy of regular norms.
Keywords: Descriptive Set Theory, Order Theory, First Periodicity Theorem, Axiom of Determinacy, Wadge hierarchy