2000/01/28 by Randall Dougherty, Dougherty, Randall, Alexander S. Kechris +1
Computer Science · Mathematics · #03D30 #03E15 (Primary) 04A15 #54H05 (Secondary) #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03D30 #msc:03E15 #msc:04A15 #msc:54H05
paper · pdf · doi:10.48550/arxiv.math/0001173
12 pages
arxiv created 2000/01/28 · openalex publication_date 2000/01/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence relations imposed by the notion of Borel reducibility. We will survey results and conjectures concerning the problem of identifying the place in this hierarchy of these equivalence relations from recursion theory and also discuss some of their implications.