Degrees of categoricity of computable structures

Degrees of Categoricity and the Hyperarithmetic Hierarchy

“…The Effective Perfect Set Theorem is the main ingredient in their proof [2]. Here, we strengthen their result and show that every degree of categoricity is hyperarithmetic.…”

“…This natural idea of a "degree of categoricity" for a computable structure was first introduced by Fokina, Kalimullin and Miller in [2].…”

Simple structures with complex symmetry

“…Here we wish to extend Proposition 2.4 to the Σ 0 n degrees, preserving the uniformity. To do so, we apply a technique of Marker and others, as given in [12] and described in [2]. For simplicity, we will assume Corollary 5.2 for the moment, so that our structures may be taken to have finite relational signature.…”

“…(Again the original predicate symbol P is in the language of A ∀ iff P A was finite or cofinite.) The usefulness of these new structures for us follows from Proposition 5.7 of [2], which we summarize here, with A ∀∃ = (A ∀ ) ∃ built by applying both of these operations in turn.…”

“…Theorem 2.5 (various authors; see [2]) In a finite signature, a structure A is ∅ (n+1) -presentable iff the structure A ∀∃ is ∅ (n) -presentable. More-over, isomorphisms between A and B correspond bijectively with isomorphisms between A ∀∃ and B ∀∃ , and if A ∀∃ and B ∀∃ are built in the natural way (as suggested by the description above), then this correspondence preserves the Turing degree of any isomorphism of degree ≥ deg(A).…”