# Bourbaki's construction of the reals

John Harrison <John.Harrison@cl.cam.ac.uk>
To: QED <qed@mcs.anl.gov>
Subject: Bourbaki's construction of the reals
In-reply-to: Your message of Thu, 22 Apr 93 09:55:39 -0400. <9304221355.AA01527@spock>
Date: Thu, 22 Apr 93 15:52:37 +0100
From: John Harrison <John.Harrison@cl.cam.ac.uk>
Message-id: <"swan.cl.cam.:047900:930422145418"@cl.cam.ac.uk>
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A peripheral point, in case the following remark by David McAllester
gives a false impression:
> The reals can be defined directly as Dedikind cuts, or defined
> indirectly by adding axioms to the concept of a topological field.
I should just emphasize that Bourbaki do not actually assert additional
axioms to those for a topological field. Rather they *construct* the
reals as the completion of the uniform space arising from the
topological group of rationals. (As David Mumford notes, a classic case
of "mathematics made difficult".)
Actually if I remember rightly (my copy of "General Topology" is at
home) that only accounts for the additive structure and they include
multiplication in some other way. There's a very beautiful treatment by
Behrend [%] which defines the (additive structure of the) positive reals
using a positional representation and recovers multiplication from the
automorphisms (i.e. identifies multiplication by x with the automorphism
which maps 1 |-> x). This allows a set of independent "axioms", as I
recall.
John Harrison (jrh@cl.cam.ac.uk).
[%] F.A. Behrend, A Contribution to the Theory of Magnitudes and the
Foundations of Analysis, Mathematische Zeitschrift, vol. 63, pp.
345-362, 1956.