On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor

Journal of Symbolic Logic:1-51 (forthcoming)
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Abstract

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify theminimalaxioms 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 oftenequivalentto the theorem over thebase theory, a weak system of ‘computable mathematics’, while most theorems are either provable in this base theory, or equivalent to one of onlyfourlogical systems. The latter plus the base theory are called the ‘Big Five’ and the associated equivalences arerobustfollowing Montalbán, i.e., stable under small variations of the theorems at hand. Working in Kohlenbach’shigher-orderRM, we obtain two new and long series of equivalences based on theorems due to Bolzano, Weierstrass, Jordan, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems. Thus, higher-order RM is much richer than its second-order cousin, boasting at least two extra ‘Big’ systems.

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Author Profiles

Sam Sanders
Ruhr-Universität Bochum
Dag Normann
University of Oslo

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References found in this work

Open questions in reverse mathematics.Antonio Montalbán - 2011 - Bulletin of Symbolic Logic 17 (3):431-454.
Pincherle's theorem in reverse mathematics and computability theory.Dag Normann & Sam Sanders - 2020 - Annals of Pure and Applied Logic 171 (5):102788.

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