On robust theorems due to Bolzano, Weierstrass, and Cantor in Reverse Mathematics

Dag Normann, Sam Sanders

Published 2021-02-09Version 1

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e. non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very often equivalent to the theorem over the base theory, a weak system of 'computable mathematics', while most theorems are either provable in the base theory, or equivalent to one of only four logical systems. The latter plus the base theory are called the 'Big Five' systems and the associated equivalences are robust, i.e. stable under small variations (of the theorems at hand). Working in Kohlenbach's higher-order RM, we obtain two long series of equivalences based on theorems due to Bolzano, Weierstrass, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems, as they are strictly intermediate between the base theory and the higher-order counterpart of weak Koenig's lemma. In this light, higher-order RM is much richer than its second-order cousin, boasting as it does two extra 'Big' systems. Our study includes numerous variations of the Bolzano-Weierstrass theorem formulated as the existence of suprema for (third-order) countable sets in Cantor space. We similarly investigate many basic theorems about the real line, limit points, and countable unions, all going back to Cantor.