Parametric Church's Thesis: Synthetic Computability without Choice
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In synthetic computability, pioneered by Richman, Bridges, and Bauer, one develops computability theory without an explicit model of computation. This is enabled by assuming an axiom equivalent to postulating a function ϕ to be universal for the space ℕ→ℕ (𝖢𝖳_ϕ, a consequence of the constructivist axiom 𝖢𝖳), Markov's principle, and at least the axiom of countable choice. Assuming 𝖢𝖳 and countable choice invalidates the law of excluded middle, thereby also invalidating classical intuitions prevalent in textbooks on computability. On the other hand, results like Rice's theorem are not provable without a form of choice. In contrast to existing work, we base our investigations in constructive type theory with a separate, impredicative universe of propositions where countable choice does not hold and thus a priori 𝖢𝖳_ϕ and the law of excluded middle seem to be consistent. We introduce various parametric strengthenings of 𝖢𝖳_ϕ, which are equivalent to assuming 𝖢𝖳_ϕ and an S^m_n operator for ϕ like in the S^m_n theorem. The strengthened axioms allow developing synthetic computability theory without choice, as demonstrated by elegant synthetic proofs of Rice's theorem. Moreover, they seem to be not in conflict with classical intuitions since they are consequences of the traditional analytic form of 𝖢𝖳. Besides explaining the novel axioms and proofs of Rice's theorem we contribute machine-checked proofs of all results in the Coq proof assistant.
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