Borel and Baire reducibility
Fundamenta Mathematicae, Tome 164 (2000) no. 1, pp. 61-69

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We prove that a Borel equivalence relation is classifiable by countable structures if and only if it is Borel reducible to a countable level of the hereditarily countable sets. We also prove the following result which was originally claimed in [FS89]: the zero density ideal of sets of natural numbers is not classifiable by countable structures.
Harvey M. Friedman. Borel and Baire reducibility. Fundamenta Mathematicae, Tome 164 (2000) no. 1, pp. 61-69. doi: 10.4064/fm-164-1-61-69
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