Extending Kleene's O Using Infinite Time Turing Machines,
Ansten M-bĂ¸rch Klev-A
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 \lambda, the supremum of the writable ordinals, while the
second extension, O^++, is proved to code a tree of height \zeta, 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 \Sigma_2 theory of L_\zeta, and by
work of Burgess that O^++ is complete with respect to the class of the
arithmetically quasi-inductive sets.
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