We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Gödel's constructible universe L. This characterization can be used to prove the generalized continuum hypothesis in L.

We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the (...) algorithmic approach to admissible recursion theory by indicating how the proof of the Sacks–Simpson theorem, i.e., the solution of Post’s problem in α-recursion theory, could be based on α-machines, without involving constructibility theory. (shrink)

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. (shrink)

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.

We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Everyset. for example, is decidable by such machines, and the semi-decidable sets form a portion of thesets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators.