TURING COMPLEXITY OF THE ORDINALS

The ordinals arise as relevant tools for measuring the strength of some construction schemes, especially in recursion theory and logic. It is a very natural idea to try to define on the ordinals the Tuning complexity classes, for instance, the notion of a polynomial time computable ordinal, etc. It is shown that this approach is irrelevant for the hierarchies below the recursive level - at least when the question is raised in the same way it has been (successfully) raised in the literature for the hierarchies above the recursive level.