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Reverse Mathematics of Real Analysis

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Reverse Mathematics is a subfield of Computability Theory and mathematical logic concerned with one main question: ``What are the necessary axioms for mathematics?" Reverse Mathematics allows us to characterize the logical strength of theorems, relating theorems across many mathematical disciplines. In this paper, we synthesize the historical work on the Reverse Math of Real Analysis by Simpson and Brown in a way that is accessible to non-logicians while a useful reference to those within the field. In particular, we describe the nuances of different definitions of closed sets, whether as complements of open sets or as sets that contain all their limit points. These distinct definitions connect to a relatively large gap of logical strength. In one case, we correct an error in the proof that "All closed sets are separably closed" implies ACA_0 over RCA_0 from Brown's 1990 paper "Notions of Closed Subsets of a Complete Separable Metric Space in Weak Subsystems of Second Order Arithmetic." We also consider how the dual definitions of closed sets affect the strength of the Baire Category Theorem. As a whole, this paper outlines a path of understanding Reverse Mathematics for those outside the field by exploring fundamental ideas from Real Analysis.
Baire Category Theorem
Closed sets
Complete separable metric spaces
Computability theory
Mathematical logic
Reverse mathematics
Ingall, Corrie Elizabeth (author)
Robinson, Stephen (committee chair)
Moore, William (committee member)
John, David (committee member)
2020-05-29T08:36:16Z (accessioned)
2020-05-29T08:36:16Z (available)
2020 (issued)
Mathematics and Statistics (discipline)
http://hdl.handle.net/10339/96869 (uri)
en (iso)
Wake Forest University
Reverse Mathematics of Real Analysis

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