Sometimes tame, sometimes wild: weak continuity

TL;DR

This paper investigates the logical strength of various weak continuity notions, revealing a spectrum from tame to wild, and connects these findings to foundational questions in mathematics and set theory.

Contribution

It classifies weak continuity notions based on their logical strength, showing some are as tame as standard continuity while others imply strong set-theoretic axioms.

Findings

Most weak continuity notions are as tame as continuity.

Seven 'wild' weak continuity notions imply strong axioms like second-order arithmetic.

Results include equivalences in higher-order Reverse Mathematics and applications to Riemann integration.

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, i.e. 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 $(\exists^3)$. Working in Kohlenbach's higher-order Reverse Mathematics, we also obtain elegant equivalences in various cases and obtain similar results for e.g. 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'.