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.
John Chisholm. Ekaterina B. Fokina. Sergey S. Goncharov. Valentina S. Harizanov. Julia F. Knight. Sara Quinn. "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