On fixed-point theorems in synthetic computability

Abstract

Lawvere's fixed point theorem captures the essence of diagonalization arguments. Cantor's theorem, Gödel's incompleteness theorem, and Tarski's undefinability of truth are all instances of the contrapositive form of the theorem. It is harder to apply the theorem directly because non-trivial examples are not easily found, in fact, none exist if excluded middle holds. We study Lawvere's fixed-point theorem in synthetic computability, which is higher-order intuitionistic logic augmented with the Axiom of Countable Choice, Markov's principle, and the Enumeration axiom, which states that there are countably many countable subsets of $\mathbb{N}$. These extra-logical principles are valid in the effective topos, as well as in any realizability topos built over Turing machines with an oracle, and suffice for an abstract axiomatic development of a computability theory. We show that every countably generated $\omega$-chain complete pointed partial order ($\omega$cppo) is countable, and that countably generated $\omega$cppos are closed under countable products. Therefore, Lawvere's fixed-point theorem applies and we obtain fixed points of all endomaps on countably generated $\omega$cppos. Similarly, the Knaster-Tarski theorem guarantees existence of least fixed points of continuous endomaps. To get the best of both theorems, namely that all endomaps on domains ($\omega$cppos generated by a countable set of compact elements) have least fixed points, we prove a synthetic version of the Myhill-Shepherdson theorem: every map from an $\omega$cpo to a domain is continuous. The proof relies on a new fixed-point theorem, the synthetic Recursion Theorem, so called because it subsumes the classic Kleene-Rogers Recursion Theorem. The Recursion Theorem takes the form of Lawvere's fixed point theorem for multi-valued endomaps.