LE3 .A278 2013

2013

Teismann, Holger

Acadia University

Bachelor of Science

Honours

Mathematics and Statistics

Mathematics & Statistics

In the typical undergraduate real analysis course, the completeness of the real num- bers is axiomatized by the existence of suprema, or more rarely by the principle of nested intervals. Additionally, the standard construction of the reals by Dedekind cuts provides an axiom based on those cuts. It is interesting to determine statements that could be taken as axioms; this is done by nding statements that hold true only when applied to the reals. In this vein, both [Teismann, 2013] and [Riemenschneider, 2001] list several statements classically known to be equivalent to completeness, as well as new results. In this thesis, we extend the e orts of these authors to collect all known complete- ness axioms while also adding several more. Additionally, we signi cantly extend a similar list, the axiomatizations of the Archimedean Property, from known results. We also explore a set of ordered elds particularly amenable to a typical undergrad- uate calculus student. Finally, we examine several statements that, at this juncture, are not known to be axioms but have not been shown otherwise.

The author retains copyright in this thesis. Any substantial copying or any other actions that exceed fair dealing or other exceptions in the Copyright Act require the permission of the author.

https://scholar.acadiau.ca/islandora/object/theses:1000