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The theorem of the means for cardinal and ordinal numbers

✍ Scribed by George Rousseau

Publisher
John Wiley and Sons
Year
1993
Tongue
English
Weight
382 KB
Volume
39
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

The theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a^2^ + b^2^ β‰₯ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality Ξ±^2^ + Ξ²^2^ β‰₯ 2Ξ±Ξ² is established and the conditions for equality are derived; stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25.

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