This book is a contribution to the general theory of equivalence relation, especially the orbit equivalence relations induced by Polish group actions. A theory is developed regarding when such equivalence relations allow countable structures considered up to isomorphism as complete invariants. This book would be of interest to mathematicians in a variety of areas.
An analog of ML-randomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π 1 1 counterparts. We prove the analogs of the Kraft-Chaitin Theorem and Schnorr's Theorem. In the new setting, while K-trivial sets exist that are not hyperarithmetical, each low for random set is. Finally, we begin to study a very strong yet effective randomness notion: Z is Π 1 1 random if Z is in no null Π 1 1 class. There is a greatest Π 1 1 null class, that is, a universal test for this notion.
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