Maximality and collapse in the hierarchy of α-c.a. degrees

Article type: Research Article

Authors: Arthur, Katherine^a | Downey, Rod^b | Greenberg, Noam^{c; *}

Affiliations: [a] School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand | [b] School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand | [c] School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

Correspondence: [*] Corresponding author. [email protected].

Abstract: In (A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (2020), Annals of Mathematics Studies, Princeton University Press), Downey and Greenberg define a transfinite hierarchy of low2 c.e. degrees – the totally α-c.a. degrees, for appropriately small ordinals α. This new hierarchy is of particular interest because it has already given rise to several natural definability results, and provides a new definable antichain in the c.e. degrees. Several levels of this hierarchy contain maximal degrees. We discuss how maximality interacts with upper cones, and the related notion of hierarchy collapse in upper cones. For example, we show that there is a totally ω-c.a. degree above which there is no maximal totally ω-c.a. degree.

Keywords: Totally α-c.a. degrees, maximal degrees, upper cones

DOI: 10.3233/COM-200297

Journal: Computability, vol. 10, no. 3, pp. 181-214, 2021