Numerical choice, Riemann integration, and Reverse Mathematics

Abstract

Riemann integration remains a well-known part of mathematics for both historical and conceptual reasons. We study basic properties like boundedness of Riemann integrable functions and related classes in mathematical logic. On one hand, weak logical systems already establish that a Riemann integrable function on the unit interval is bounded or dominated by a continuous function. On the other hand, the following slight generalisation already implies a rather strong logical system, namely the `Big Five' system ATR$_{0}$ which accommodates transfinite recursion. $$ \text{For $f:\mathbb{R}\rightarrow \mathbb{R}$ Riemann integrable on any interval $[-a, a]$ for $a>0$, there is continuous $g:\mathbb{R}\rightarrow \mathbb{R}$ with $f(x)\leq g(x)$ for all $x\in \mathbb{R}$.} $$ As part of the \emph{Reverse Mathematics} program, we obtain equivalences for the centred statement and variations involving the axiom of numerical choice. A central result is that numerical choice for $Π_{1}^{1}$-formulas is equivalent to ATR$_{0}$. We also obtain equivalences involving basic properties of metric spaces and establish connections to Kohlenbach's generalisations of weak König's lemma, Cousin's lemma, and the representation of open sets.