Classes of structures with universe a subset of omega(1)
We continue recent work on computable structure theory in the setting of omega(1). 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 Sigma(1)(1) equivalence relations on omega. Our equivalence relations are on omega(1). In the standard setting, Sigma(1)(1) sets are characterized in terms of paths through trees. In the setting of omega(1), we use a new characterization of Sigma(1)(1) sets that involves clubs in omega(1). Finally, we present some new results about omega(1)-computable categoricity for fields.
omega(1)-computability computable uncountable structure turing computable embeddings computable categoricity complete equivalence relation