A computably enumerable many-one degree with no least finite-one degree

TL;DR

This paper constructs a nonrecursive computably enumerable set whose many-one degree lacks a least finite-one degree, answering a long-standing question negatively within computably enumerable degrees.

Contribution

It provides the first explicit example of such a degree, using a finite-injury priority construction with novel techniques like virtual target sets and dynamic traps.

Findings

Constructed a nonrecursive c.e. set with no least finite-one degree.

Demonstrated that the many-one degree of this set contains no minimal finite-one degree.

Solved a previously open problem in computability theory.

Abstract

Richter, Stephan, and Zhang asked whether every nonrecursive many-one degree contains a least finite-one degree. We solve this question in the negative, already within the class of computably enumerable many-one degrees. Positive answers are known in two disjoint natural settings: for a measure-one and comeager class of $m$-rigid sets, and, in a companion paper, for computably enumerable many-one degrees containing a $D$-maximal set. We construct a nonrecursive \ce\ set $A$ such that for every set $X \eqm A$ there exists a c.e.\ set $B \eqm A$ with $X \not\lfo B$. Hence the many-one degree of $A$ contains no least finite-one degree. The proof is a finite-injury priority construction based on virtual target sets and a dynamic trap mechanism forcing any putative finite-one reduction either to violate finite-oneness or to compute an incorrect reduction.