Degrees That Are Not Degrees of Categoricity

A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B ∼= A there is an isomorphism f : B → A with f ≤T x. A degree x is a degree of categoricity if there is a computable structure A such that A is x-computably categorical, and for all y, if A is y-computably categorical then x ≤T y. We construct a Σ2 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.