A Lambda Calculus for Real Analysis

Abstract Stone Duality is a revolutionary theory that works directly with computable continuous functions, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, but the reasoning looks remarkably like a sanitised form of that in classical topology. This paper is an introduction to ASD for the general mathematician, and applies it to elementary real analysis. It culminates in the Intermediate Value Theorem, i.e. the solution of equations fx = 0 for continuous f : R → R. As is well known from both numerical and constructive considerations, the equation cannot be solved if f “hovers” near 0, whilst tangential solutions will never be found. In ASD, both of these failures and the general method of finding solutions of the equation when they exist are explained by the new concept of “overtness”. The zeroes are captured, not as a set, but by higher-type operators and ♦ that remain (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than sets of points leads to a very closely dual treatment of open and closed subspaces, without the double negations of intuitionistic approaches. In this, the dual of compactness is overtness, and whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD. Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis and recursive enumerability in recursion theory. As a further application of connectedness, we also show that every open set of the line is uniquely expressible as a countable union of intervals, in a suitable constructive sense, which is not the case in Bishop’s theory.