Notre Dame Journal of Formal Logic

Infinite Time Decidable Equivalence Relation Theory

Samuel Coskey and Joel David Hamkins


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.

Article information

Notre Dame J. Formal Logic, Volume 52, Number 2 (2011), 203-228.

First available in Project Euclid: 28 April 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D30: Other degrees and reducibilities 03D65: Higher-type and set recursion theory 03E15: Descriptive set theory [See also 28A05, 54H05]

set theory descriptive set theory infinite time computation equivalence relations


Coskey, Samuel; Hamkins, Joel David. Infinite Time Decidable Equivalence Relation Theory. Notre Dame J. Formal Logic 52 (2011), no. 2, 203--228. doi:10.1215/00294527-1306199.

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