# The Reals as Rational Cauchy Filters

### From NZJM

**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 (where
the Cauchy condition is defined in terms of the absolute value function
on ) 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.