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  1.  25
    New jump operators on equivalence relations.John D. Clemens & Samuel Coskey - 2022 - Journal of Mathematical Logic 22 (3).
    We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group [Formula: see text] we introduce the [Formula: see text]-jump. We study the elementary properties of the [Formula: see text]-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups [Formula: see text], the [Formula: see text]-jump is proper in the sense that for any Borel equivalence relation [Formula: see text] the [Formula: see (...)
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  2.  21
    Computable Reducibility of Equivalence Relations and an Effective Jump Operator.John D. Clemens, Samuel Coskey & Gianni Krakoff - forthcoming - Journal of Symbolic Logic:1-22.
    We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).
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  3.  60
    Polish metric spaces: Their classification and isometry groups.John D. Clemens, Su Gao & Alexander S. Kechris - 2001 - Bulletin of Symbolic Logic 7 (3):361-375.
    § 1. Introduction. In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space.Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks : “The classification of Polish spaces up to (...)
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  4.  37
    Isomorphism of Homogeneous Structures.John D. Clemens - 2009 - Notre Dame Journal of Formal Logic 50 (1):1-22.
    We consider the complexity of the isomorphism relation on countable first-order structures with transitive automorphism groups. We use the theory of Borel reducibility of equivalence relations to show that the isomorphism problem for vertex-transitive graphs is as complicated as the isomorphism problem for arbitrary graphs and determine for which first-order languages the isomorphism problem for transitive countable structures is as complicated as it is for arbitrary countable structures. We then use these results to characterize the complexity of the isometry relation (...)
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  5.  24
    Isometry of Polish metric spaces.John D. Clemens - 2012 - Annals of Pure and Applied Logic 163 (9):1196-1209.
  6.  25
    Weakly pointed trees and partial injections.John D. Clemens - 2008 - Journal of Symbolic Logic 73 (1):363-368.
    We define the notion of a weakly pointed tree, and characterize the amount of genericity necessary to prevent a uniformly branching tree being weakly pointed. We use these ideas to show there is no topological analogue of a measure-theoretic selection theorem of Graf and Mauldin.
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  7.  22
    Classifying Borel Automorphisms.John D. Clemens - 2007 - Journal of Symbolic Logic 72 (4):1081 - 1092.