  Turing's famous `machine' model constitutes the first intuitively convincing
framework for computing with real numbers. Kleene's computation schemes S1-S9
extend Turing's approach to computing with objects of any finite type. Both
frameworks have their pros and cons and it is a natural question if there is an
approach that marries the best of both the Turing and Kleene worlds. In answer
to this question, we propose a considerable extension of the scope of Turing's
approach. Central is a fragment of the Axiom of Choice involving continuous
choice functions, going back to Kreisel-Troelstra and intuitionistic analysis.
Put another way, we formulate a relation `is computationally stronger than'
involving third-order objects that overcomes (many of) the pitfalls of the
Turing and Kleene frameworks.
