Preserving Categoricity and Complexity of Relations

Abstract

In [3, 6, 7], it was proved that for each computable ordinal α, there is a structure that is Δ0 α categorical but not relatively Δ0 α categorical. The original examples were not familiar algebraic kinds of structures. In [11], it was shown that for α = 1, there are further examples in several familiar classes of structures, including rings and 2-step nilpotent groups. Similar example for all computable successor ordinals were constructed in [12]. In the present paper, this result is extended to computable limit ordinals. We know of an example of an algebraic field that is computably categorical but not relatively computably categorical. Here we show that for each computable limit ordinal α > ω, there is a field which is Δ0 α categorical but not relatively Δ0 α categorical. Examples on dimension and complexity of relations are given.