2019/10/17 by Shani, Assaf
#03E15 #03E25 #03E47 #03E75 #37A20 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1910.08188
This paper deals with countable products of countable Borel equivalence relations and equivalence relations "just above" those in the Borel reducibility hierarchy. We show that if E is strongly ergodic with respect to μ then E^ℕ is strongly ergodic with respect to μ^ℕ. We answer questions of Clemens and Coskey regarding their recently defined Γ-jump operations, in particular showing that the ℤ2-jump of E_∞ is strictly above the ℤ-jump of E_∞. We study a notion of equivalence relations which can be classified by infinite sequences of "definably countable sets". In particular, we define an interesting example of such equivalence relation which is strictly above E_∞^ℕ, strictly below =+, and is incomparable with the Γ-jumps of countable equivalence relations. We establish a characterization of strong ergodicity between Borel equivalence relations in terms of symmetric models. The proofs then rely on a fine analysis of the very weak choice principles "every sequence of E-classes admits a choice sequence", for various countable Borel equivalence relations E.