Between Turing and Kleene

Sam Sanders

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arXiv · 2111.05052

Between Turing and Kleene

Sam Sanders

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Abstract

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.

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BibTeX RIS

Sam Sanders. 2021-11-09. Between Turing and Kleene. https://arxiv.org/abs/2111.05052

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