2010
DOI: 10.1016/j.apal.2009.10.002
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The effective theory of Borel equivalence relations

Abstract: The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and Harrington-Kechris-Louveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on P(ω). In this article we examine the effective content … Show more

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Cited by 23 publications
(33 citation statements)
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“…Theorem 3.5 does not have an effective version because determinacy is not guaranteed in the effective world. As pointed to us by an anonymous referee, the arguments given by Fokina et al (2010) yield the following result. Proof.…”
Section: (Hardness Theorem)mentioning
confidence: 86%
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“…Theorem 3.5 does not have an effective version because determinacy is not guaranteed in the effective world. As pointed to us by an anonymous referee, the arguments given by Fokina et al (2010) yield the following result. Proof.…”
Section: (Hardness Theorem)mentioning
confidence: 86%
“…Proof. We follow the arguments in the proof of Theorem 9 in Fokina et al (2010). Their Theorem 6 establishes that there is a uniform sequence (A n ) n∈N of nonempty Π 0 1 subsets of N ω such that, for each n, there is no hyperarithmetical function…”
Section: (Hardness Theorem)mentioning
confidence: 99%
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“…The theory of effective equivalence relations has grown to a rather broad area; we cite [1,18,9] for recent results on this subject. Many results of this paper can be stated in terms of Δ 0 2 -embeddings between effectively presented equivalence structures.…”
Section: Effectively Presentable Equivalence Structuresmentioning
confidence: 99%
“…The complexity of the isomorphism relation on various classes of countable structures was measured using the idea of effective transformations. In the recent work [11] the general theory of effectively Borel (i.e., ∆ 1 1 ) equivalence relations on effectively presented Polish spaces was developed via the notion of effective Borel reducibility. The resulting structure turned out to be much more complex than in the classical case.…”
Section: Introductionmentioning
confidence: 99%