Ordinal machines and admissible recursion theory

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.