Logic Seminar

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There is an array of long-standing open problems in the theory of countable Borel equivalence relations (CBER), all of which state that the class of hyperfinite CBERs is nice in some way. For instance, the unresolved Union Problem asks whether the class of hyperfinite CBERs is closed under increasing unions, and in a different direction, it is also open whether the hyperfinite CBERs form a Π11 set, which would be nicer than the naive complexity of Σ12. There are many other such problems, and it is widely believed that if one of them is false, then most of the others will be false as well, although there is no formal statement to this effect. To this end, we show an implication between the two aforementioned problems: precisely, we show that if the Union Problem has a negative answer, then the Borel complexity of the class of hyperfinite CBERs is as high as possible, namely Σ12-complete. This is joint with Joshua Frisch and Zoltan Vidnyanszky.