Open Access
2011 Infinite Time Decidable Equivalence Relation Theory
Samuel Coskey, Joel David Hamkins
Notre Dame J. Formal Logic 52(2): 203-228 (2011). DOI: 10.1215/00294527-1306199


We introduce an analogue of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time computable functions. Many basic aspects of the classical theory remain intact, with the added bonus that it becomes sensible to study some special equivalence relations whose complexity is beyond Borel or even analytic. We also introduce an infinite time generalization of the countable Borel equivalence relations, a key subclass of the Borel equivalence relations, and again show that several key properties carry over to the larger class. Lastly, we collect together several results from the literature regarding Borel reducibility which apply also to absolutely Δ 2 1 reductions, and hence to the infinite time computable reductions.


Download Citation

Samuel Coskey. Joel David Hamkins. "Infinite Time Decidable Equivalence Relation Theory." Notre Dame J. Formal Logic 52 (2) 203 - 228, 2011.


Published: 2011
First available in Project Euclid: 28 April 2011

zbMATH: 1233.03050
MathSciNet: MR2794652
Digital Object Identifier: 10.1215/00294527-1306199

Primary: 03D30 , 03D65 , 03E15

Keywords: descriptive set theory , equivalence relations , infinite time computation , Set theory

Rights: Copyright © 2011 University of Notre Dame

Vol.52 • No. 2 • 2011
Back to Top