The Reals as Rational Cauchy Filters


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New Zealand Journal of Mathematics

Vol. 46, (2016), Pages 21-51

Ittay Weiss

School of Computing,

Information and Mathematical Sciences,

The University of the South Pacific,

Suva, Fiji.

Abstract We present, alongside a historical note on the development of the study of the real numbers, a detailed and elementary construction of the real numbers from the rational numbers a la Bourbaki. The real numbers are defined to be the set of all minimal Cauchy filters in \mathbb{Q} (where the Cauchy condition is defined in terms of the absolute value function on \mathbb{Q}) and are proven directly, without employing any of the techniques of uniform spaces, to form a complete ordered field. The construction can be seen as a variant of Bachmann's construction by means of nested rational intervals, allowing for a canonical choice of representatives.

Keywords real numbers, construction of the reals, filter, Cauchy filter, minimal Cauchy filter, rational filter, historical survey of the real numbers, criticism of Dedekind cuts, criticism of Cauchy's construction.

Classification (MSC2000) 00A05, 01A55, 01A60, 97I99.

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