Degrees that are not degrees of categoricity

TL;DR

This paper investigates the concept of degrees of categoricity in computable structures, constructing examples and identifying classes of degrees that are not degrees of categoricity, including certain noncomputable and hyperimmune-free degrees.

Contribution

It constructs a Sigma_2 set with a degree not of categoricity and identifies broad classes of degrees that lack this property, advancing understanding of computable structure categoricity.

Findings

Constructed a Sigma_2 set with a non-categoricity degree.

Showed that degrees of 2-generic sets relative to perfect trees are not degrees of categoricity.

Proved that all noncomputable hyperimmune-free degrees are not degrees of categoricity.

Abstract

A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B isomorphic to A there is an isomorphism f:B -> A with f computable in x. A degree x is a degree of categoricity if there is a computable A such that A is x-computably categorical, and for all y, if A is y-computably categorical then y computes x. We construct a Sigma_2 set whose degree is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity.