Functionary Models of Real Analysis

Abstract

A real number can have more than one name: $0.5$ and $0.4999\ldots$ denote the same thing, and which numbers enjoy such a doubling depends on the base one writes in. We make that dependence the object of study. Fixing a system function $\vartheta$, which assigns a base $\vartheta(n)\geq2$ to every position independently, we build the real numbers as equivalence classes of digit functions $f:\omega\rightarrow\omega$, where the equivalence is generated by two local carrying moves, contraction and broadening, and tested by agreement on finite initial segments. Each class turns out to contain at most two canonical representatives (up to a sign representation), its primary and secondary auxiliary functions, so the doubling above is a theorem of the theory rather than a convention imposed on it. We define order, addition and multiplication and verify the axioms of a Dedekind-complete ordered field, so that by categoricity every choice of $\vartheta$ delivers $\mathbb{R}$ itself. The models are therefore indistinguishable as ordered fields and differ only in how their elements are named, and since $\vartheta$ ranges over an uncountable parameter space, that naming can be chosen to suit a problem. We show what this buys: in any base whose partial products absorb every denominator, a non-zero real number is rational precisely when it has a second name, and Cantor's 1869 irrationality criterion for Cantor series follows from the representation theory rather than from number theory.