Extending Kleene’s O Using Infinite Time Turing Machines, or How With Time She Grew Taller and Fatter.

Abstract

We define two successive extensions of Kleene’s O using infinite time Turing machines. The first extension, O +, is proved to code a tree of height λ, the supremum of the writable ordinals, while the second extension, O ++, is proved to code a tree of height ζ, the supremum of the eventually writable ordinals. Furthermore, we show that O + is computably isomorphic to h, the lightface halting problem of infinite time Turing machine computability, and that O ++ is computably isomorphic to s, the set of programs that eventually write a real. The last of these results implies, by work of Welch, that O ++ is computably isomorphic to the Σ2 theory of Lζ, and, by work of Burgess, that O ++ is complete with respect to the class of the arithmetically quasi-inductive sets. This leads us to conjecture the existence of a parallel of hyperarithmetic theory at the level of Σ2(Lζ), a theory in which O ++ plays the role of O, the arithmetically quasi-inductive sets play the role of Π1 1, and the eventually writable reals play the role of ∆1 1.