TL;DR
This paper explores the equivalence of many real analysis theorems to Dedekind completeness, examining less-known ordered fields and discussing their implications for understanding and teaching the real numbers.
Contribution
It demonstrates how various theorems serve as alternative completeness axioms and provides historical and pedagogical insights into ordered fields.
Findings
Many theorems are equivalent to Dedekind completeness
Provides a survey of less-familiar ordered fields
Discusses implications for teaching real analysis
Abstract
Many of the theorems of real analysis, against the background of the ordered field axioms, are equivalent to Dedekind completeness, and hence can serve as completeness axioms for the reals. In the course of demonstrating this, the article offers a tour of some less-familiar ordered fields, provides some of the relevant history, and considers pedagogical implications.
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