Reals a la Eudoxus (updated from re: Tate Reals)

Reals a la Eudoxus (updated from re: Tate Reals)

[Stephen H. Schanuel]

My 'construction' of the reals, referred to by Ross and Mike, was
suggested by an observation of Tate about bilinear maps many years
earlier, but essentially goes back to Eudoxus, who noted that to measure
the ratio of a line segment to another with any desired accuracy, it isn't
necessary to cut either of them up--instead multiply the 'numerator'
segment by a large integer--thus pointing the way to a direct construction
of the reals as ring from the integers as mere additive group.
(Incidentally, I take the task to be relating the continuous to the
discrete, rather than constructing the former from the latter, but never
mind.) R is the ring of endomorphisms E(T(L))of the group T(L) of
translations (rigid maps at finite distance from the identity) of the line
L. If we start instead with a discrete line D (a line of dots), then T(D)
is isomorphic to Z (additive group without preferred generator). E(T(D) is
the ring Z, but instead take A(T(D)), 'almost homomorphisms' (as in Mike's
description) from Z to Z. A(T(D)) is an additive goup with a
'multiplication' by composition, but not a ring, since one distributive
law and commutativity of multiplication fail; but A(T(D)) modulo bounded
maps is R. The maps in both directions are easy: send the real r to the
map 'multiply by r and round down', and send the almost homomorphism f to
the limit of the Cauchy sequence f(n)/n.

Steve Schanuel

Re: Reals a la Eudoxus (updated from re: Tate Reals)

[Michael Barr]

I don't want to be disagreeable, but it seems clear to me that this
construction gives Cauchy reals, not Dedekind reals and only the latter
can be said to go back to Eudoxus.  Indeed, given a nearly function f,
with bound B on the near linearity, it is easy to see that |(fn/n) -
(fm/m)| =< B(1/n + 1/m) so that the sequence fn/n is Cauchy.

Michael