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ON ROBUST THEOREMS DUE TO BOLZANO, WEIERSTRASS, JORDAN, AND CANTOR

Part of: General logic Computability and recursion theory Proof theory and constructive mathematics

Published online by Cambridge University Press:  03 October 2022

DAG NORMANN  and
SAM SANDERS
DAG NORMANN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSLO OSLO, NORWAY E-mail: dnormann@math.uio.no
SAM SANDERS*
Affiliation:
INSTITUTE FOR PHILOSOPHY II RUB BOCHUM BOCHUM, GERMANY
*
E-mail: sasander@me.com
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Abstract

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 this base theory, or equivalent to one of only four logical systems. The latter plus the base theory are called the ‘Big Five’ and the associated equivalences are robust following Montalbán, i.e., stable under small variations of the theorems at hand. Working in Kohlenbach’s higher-order RM, 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.

Keywords

countable setsBolzano–Weierstrass theoremReverse Mathematicshigher-order computabilityKleene’s S1–S9bounded variationregulated functions

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03F35: Second- and higher-order arithmetic and fragments 03D55: Hierarchies 03D30: Other degrees and reducibilities

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 3 , September 2024 , pp. 1077 - 1127
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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