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Axiom of Choice Equivalences and Some Applications

In this paper several equivalences of the axiom of choice are examined. In particular, the axiom of choice, Zorn's lemma, Tukey's lemma, the Hausdorff maximal principle, and the well-ordering theorem are shown to be equivalent. Cardinal and ordinal number theory is also studied. The Schroder-Bernstein theorem is proven and used in establishing order results for cardinal numbers. It is also demonstrated that the first uncountable ordinal space is unique up to order isomorphism. We conclude by encountering several applications of the axiom of choice. In particular, we show that every vector space must have a Hamel basis and that any two Hamel bases for the same space must have the same cardinality. We establish that the Tychonoff product theorem implies the axiom of choice and see the use of the axiom of choice in the proof of the Hahn- Banach theorem.

Identiferoai:union.ndltd.org:unt.edu/info:ark/67531/metadc504170
Date08 1900
CreatorsRace, Denise T. (Denise Tatsch)
ContributorsLewis, Paul Weldon, Hagan, Melvin R.
PublisherNorth Texas State University
Source SetsUniversity of North Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis or Dissertation
Formatiii, 60 leaves, Text
RightsPublic, Race, Denise T. (Denise Tatsch), Copyright, Copyright is held by the author, unless otherwise noted. All rights reserved.

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