Archive for Mathematical Logic

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

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

  • Ansten Mørch Klev
Open Access


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.


Infinite time Turing machines Ordinal notation systems 

Mathematics Subject Classification (2000)

03D10 03D70 03F15 



The results presented in this paper are taken from the author’s MSc thesis [6]. The author wishes to express his sincerest thanks to Joel David Hamkins for very kind and helpful supervision.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of PhilosophyMcGill UniversityMontréalCanada

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