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.
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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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Klev, A.M. Infinite time extensions of Kleene’s \({\mathcal{O}}\) . Arch. Math. Logic 48, 691–703 (2009). https://doi.org/10.1007/s00153-009-0146-2
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DOI: https://doi.org/10.1007/s00153-009-0146-2