Investigating the Computable Friedman–stanley Jump

Abstract

The Friedman–Stanley jump, extensively studied by descriptive set theorists, is a fundamental tool for gauging the complexity of Borel isomorphism relations. This paper focuses on a natural computable analog of this jump operator for equivalence relations on $\omega $ , written ${\dotplus }$ , recently introduced by Clemens, Coskey, and Krakoff. We offer a thorough analysis of the computable Friedman–Stanley jump and its connections with the hierarchy of countable equivalence relations under the computable reducibility $\leq _c$ . In particular, we show that this jump gives benchmark equivalence relations going up the hyperarithmetic hierarchy and we unveil the complicated highness hierarchy that arises from ${\dotplus }$ .