Archive for Mathematical Logic

, Volume 48, Issue 7, pp 691–703 | Cite as

Infinite time extensions of Kleene’s \({\mathcal{O}}\)

Open Access
Article

Abstract

Using infinite time Turing machines we define two successive extensions of Kleene’s \({\mathcal{O}}\) and characterize both their height and their complexity. Specifically, we first prove that the one extension—which we will call \({\mathcal{O}^{+}}\)—has height equal to the supremum of the writable ordinals, and that the other extension—which we will call \({\mathcal{O}}^{++}\)—has height equal to the supremum of the eventually writable ordinals. Next we prove that \({\mathcal{O}^+}\) is Turing computably isomorphic to the halting problem of infinite time Turing computability, and that \({\mathcal{O}^{++}}\) is Turing computably isomorphic to the halting problem of eventual computability.

Keywords

Infinite time Turing machines Ordinal notation systems 

Mathematics Subject Classification (2000)

03D10 03D70 03F15 

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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of PhilosophyMcGill UniversityMontréalCanada

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