Borel-amenable Reducibilities for Sets of Reals

TL;DR

Under the Axiom of Determinacy, the paper demonstrates that hierarchies induced by well-behaved Borel function classes resemble the Wadge hierarchy, extending understanding of reducibility structures on sets of reals.

Contribution

The paper introduces a framework for Borel-amenable reducibilities, showing they produce hierarchies similar to the Wadge hierarchy under the Axiom of Determinacy.

Findings

Hierarchies mirror the Wadge hierarchy structure

Well-behaved Borel function classes induce similar degrees

Results depend on the Axiom of Determinacy

Abstract

We show that if $\mathcal{F}$ is any "well-behaved" subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on $\pow(\mathbb{R})$ induced by $\mathcal{F}$ turns out to look like the Wadge hierarchy (which is the special case where $\mathcal{F}$ is the set of continuous functions).