We show that for every computable limit ordinal α, there is a computable structure 𝒜 that is Δα⁰ categorical, but not relatively Δα⁰ categorical (equivalently, it does not have a formally Σα⁰ 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 Σα⁰ on 𝒜, but not relatively intrinsically Σα⁰ on 𝒜 (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in , , and  establish the same facts for computable successor ordinals α.
"Intrinsic bounds on complexity and definability at limit levels." J. Symbolic Logic 74 (3) 1047 - 1060, September 2009. https://doi.org/10.2178/jsl/1245158098