We show that for every computable limit ordinal α ,t here is ac omputable structure A that is ∆ 0 categorical, but not relatively ∆ 0 categorical, i.e., does not have a formally Σ 0 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 0 on A, but not relatively intrinsically Σ 0 on A, i.e., not definable by a computable Σα formula with
['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