Church’s Theorem and Randomness

Abstract

Alonzo Church demonstrated in a logically rigorous way that Leibniz’s dream of a logical calculus that could decide all truths was not only unfulfilled but unfulfillable. According to Church’s Theorem (1936a), even firstorder logic is undecidable. Kleene’s (1936) proof of the equivalence of the ChurchKleene notion of λ-definability and the Herbrand-Gödel (1931) notion of general recursiveness and (1937) proof of the equivalence of λ-definability and computability were regarded by Church as evidence for Church’s Thesis (1936). Church’s Thesis is the philosophical claim that a function of positive integers is effectively calculable if and only if it is λ-definable.¹ Using the notion of effective calculability, (1940) proposed a definition of randomness. Church’s definition can be seen as a precursor to algorithmic definitions of randomness formulated independently by (1963), 1965) and (1966). In this paper we sketch one way of relating Church’s Theorem to randomness by way of a recent generalization of the’paradox of the Liar (Mar and Grim 1991) that uses the mathematics of chaos.