Some results on the Generalized Weihrauch Hierarchy
Gian Marco Osso
Abstract:
Weihrauch degree theory is a field of study which attempts to classify mathematical theorems based on their computational content. Brattka and Gherardi obtained a picture of the Weihrauch degrees of theorems of mathematical analysis which is stratified by the so-called choice and boundedness principles. We generalise their work to the realm of higher descriptive set theory, where the role of Baire space ω ω is taken by generalized Baire space κκ (for uncountable cardinals κ satisfying κ<κ = κ) and the role of the real line is taken by Galeotti’s generalized real line Rκ . To achieve this, we adapt a framework of Brattka to obtain a general theory of κ-computable metric spaces. Subsequenty, we check which of the techniques for proving non-reductions in the classical setting can be transported to the generalized setting. Lastly, we draw from the classical literature to prove several Weihrauch reducibility results in the generalized context. The result is a fairly complete picture (page 97) of the Weihrauch degrees of many of the currently known generalized analysis theorems and of the generalized choice and boundedness principles.