We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of F F -reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ω.
We develop a version of Cichoń's diagram for cardinal invariants on the generalized Cantor space 2 κ or the generalized Baire space κ κ where κ is an uncountable regular cardinal. For strongly inaccessible κ, many of the ZFC-results about the order relationship of the cardinal invariants which hold for ω generalize; for example we obtain a natural generalization of the Bartoszyński-Raisonnier-Stern Theorem. We also prove a number of independence results, both with < κ-support iterations and κ-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.
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