Sometimes tame, sometimes wild: Weak continuity

Abstract

Continuity is one of the most central notions in mathematics, physics and computer science. An interesting associated topic is decompositions of continuity, where continuity is shown to be equivalent to the combination of two or more weak continuity notions. In this paper, we study the logical properties of basic theorems about weakly continuous functions, like the supremum principle for the unit interval. We establish that most weak continuity notions are as tame as continuity, that is, the supremum principle can be proved from the relatively weak arithmetical comprehension axiom only. By contrast, for seven ‘wild’ weak continuity notions, the associated supremum principle yields rather strong axioms, including Feferman's projection principle, full second-order arithmetic or Kleene's associated quantifier $\left({\exists }^3\right)$. Working in Kohlenbach's higher-order Reverse Mathematics, we also obtain elegant equivalences in various cases and obtain similar results for for example, Riemann integration. We believe these results to be of interest to mainstream mathematics as they cast new light on the distinction of ‘ordinary mathematics’ versus ‘foundations of mathematics/set theory’.