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Published online by Cambridge University Press: 12 March 2014
We show that for every computable limit ordinal α, there is a computable structure that is
categorical, but not relatively
categorical (equivalently, it does not have a formally
Scott family). We also show that for every computable limit ordinal α, there is a computable structure
with an additional relation R that is intrinsically
on
, but not relatively intrinsically
on
(equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.