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November 2020 Reverse Mathematics of Topology: Dimension, Paracompactness, and Splittings
Sam Sanders
Notre Dame J. Formal Logic 61(4): 537-559 (November 2020). DOI: 10.1215/00294527-2020-0021

Abstract

Reverse mathematics (RM hereafter) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. As to splittings, there are some examples in RM of theorems A , B , C such that A ( B C ) , that is, A can be split into two independent (fairly natural) parts B and C , and the aforementioned topological notions give rise to a number of splittings involving highly natural A , B , C . Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of, for example, the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory RCA 0 . We obtain similarly “exceptional” results for the Urysohn identity, the Lindelöf lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice.

Citation

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Sam Sanders. "Reverse Mathematics of Topology: Dimension, Paracompactness, and Splittings." Notre Dame J. Formal Logic 61 (4) 537 - 559, November 2020. https://doi.org/10.1215/00294527-2020-0021

Information

Received: 29 July 2019; Accepted: 27 May 2020; Published: November 2020
First available in Project Euclid: 29 December 2020

Digital Object Identifier: 10.1215/00294527-2020-0021

Subjects:
Primary: 03B30
Secondary: 03F35

Rights: Copyright © 2020 University of Notre Dame

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Vol.61 • No. 4 • November 2020
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