Infinite time extensions of Kleene’s \({\mathcal{O}}\)
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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 systemsMathematics Subject Classification (2000)
03D10 03D70 03F15Notes
Acknowledgments
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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