Intrinsic bounds on complexity and definability at limit levels

Abstract

We show that for every computable limit ordinal α, there is a computable structure 𝒜 that is Δ_α⁰ categorical, but not relatively Δ_α⁰ categorical (equivalently, it does not have a formally Σ_α⁰ Scott family). We also show that for every computable limit ordinal α, there is a computable structure 𝒜 with an additional relation R that is intrinsically Σ_α⁰ on 𝒜, but not relatively intrinsically Σ_α⁰ on 𝒜 (equivalently, it is not definable by a computable Σ_α formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.