Re: Axioms for positive elements of an ordered field?

In article <193lj1p96ete1d76p78h6pehk06gva216l@xxxxxxx> ullrich@xxxxxxxxxxxxxxxx writes:
> On Wed, 28 Sep 2005 01:32:36 GMT, "*** T. Winter" <***.Winter@xxxxxx>
> wrote:
>
> >In article <86p64smrf9z.fsf@xxxxxxxxxxxxxxxxxx> Esa A E Peuha <esa.peuha@xxxxxxxxxxx> writes:
> > > "*** T. Winter" <***.Winter@xxxxxx> writes:
> > >[...]
> >
> >Closer to the "L" definition is what Baudet did (Dutch mathematician,
> >1891-1921). He started with arbitrary sets of non-negative rationals
> >(why non-negative I do not know).
>
> Seems like a reasonable guess would be because of the problem
> that led to this thread in the first place - if we only talk
> about non-negative reals in terms of sets of non-negative
> rationals that greatly simplifies the definition of the product
> of two reals. ???

I think so, yes. And now, when I reread various parts of the book by Schuh,
I find that of the four methods described (Cauchy, Dedekind, Weierstrass
and Baudet), only the method of Cauchy is described by starting with all
rationals. The other methods are described starting with the non-negative
rationals. But I have found the original article by Baudet about it.
<http://www.ru.nl/w-en-s/gmfw/bronnen/pbaudet4.html> (in Dutch).
There he actually uses Dedekinds R sets. But his final remark is
revealing (in translation):
In para 2 the real number is actually introduced as the lower bound of
a set of rational numbers >= 0. Of course one can build the theory with
as base the upper bound. In the second section "greater" should be
replaced by "smaller". Moreover, one has to add that each collection
of numbers should have an upperbound. As that is not needed in this
version, the current one is a bit simpler.
One could also introduce the real numbers after the rationals have been
expanded with the negative rationals. In that case sets of numbers
should be thought of as bounded on one side. The downside of this latter
order of introduction is that (with this theory as well as with the
theory from Dedekind), the multiplication is much more cumbersome, so
that we lose elegance.
And I think he is was right. Starting with the non-negative rationals
and R sets gives the simplest introduction of the reals.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.