Item Details

abstract

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.

subject

Baire Category Theorem

Closed sets

Complete separable metric spaces

Computability theory

Mathematical logic

Reverse mathematics

contributor

Ingall, Corrie Elizabeth (author)

Robinson, Stephen (committee chair)

Moore, William (committee member)

John, David (committee member)

date

2020-05-29T08:36:16Z (accessioned)

2020-05-29T08:36:16Z (available)

2020 (issued)

degree

Mathematics and Statistics (discipline)

identifier

http://hdl.handle.net/10339/96869 (uri)

language

en (iso)

publisher

Wake Forest University

title

Reverse Mathematics of Real Analysis

type

Thesis