Cantor–Bendixson ranks of effectively closed sets in admissible recursion theory

Article type: Research Article

Authors: Johnston, Reese^{; *; **}

Affiliations: Robinson Center for Young Scholars, University of Washington, Seattle, WA, United States. reesej2@uw.edu

Correspondence: [**] Corresponding author. reesej2@uw.edu.

Note: [*] The author would like to thank Yang Yue at the National University of Singapore for a series of conversations around the Cantor–Bendixson derivative that assisted greatly in sharpening the ideas in this paper.

Abstract: The range of possible ordinal values for the Cantor–Bendixson rank of an effectively closed set in Cantor space, and the relationship between the ranks and the Turing degrees of its members, have been closely studied. The wide array of results on this topic have often made use of useful topological features of Cantor space (notably, compactness) and of key properties of computable well-orders. In this paper, a number of these results are extended to the case of Cantor space on the first uncountable ordinal, in which these properties do not hold, using admissible recursion theory.

Keywords: Computability, admissible recursion theory, point-set topology

DOI: 10.3233/COM-180187

Journal: Computability, vol. 9, no. 2, pp. 111-125, 2020