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

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.