Classes of structures with universe a subset of ω1

We continue recent work on computable structure theory in the setting of ω1. We prove the analogue of a result from Fokina et al. (2012 J. Symbolic Logic, 77, 122–132) saying that isomorphism of computable structures lies ‘on top’ among � 1 equivalence relations on ω. Our equivalence relations are on ω1. In the standard setting, � 1 1 sets are characterized in terms of paths through trees. In the setting of ω1, we use a new characterization of � 1 1 sets that involves clubs in ω1. Finally, we present some new results about ω1-computable categoricity for fields.

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