Re: Axioms for positive elements of an ordered field?

In article <86p64smrf9z.fsf@xxxxxxxxxxxxxxxxxx> Esa A E Peuha <esa.peuha@xxxxxxxxxxx> writes:
> "*** T. Winter" <***.Winter@xxxxxx> writes:
>
> > In article <dntfj1dcobfj8aulrnd86ra1q8dcuahj5p@xxxxxxx> ullrich@xxxxxxxxxxxxxxxx writes:
> > > I wouldn't know about the history; if so then one wonders
> > > why they're called Dedekind cuts. (Ok, a more common
> > > definition of "Dedekind cut" involves a pair of sets
> > > of rationals, but there's no real difference.)
> >
> > But that is indeed the difference between Dedekind cuts and Weierstrass'
> > definition.
>
> Really? I thought Weierstrass defined real numbers so that eg. pi would
> be the set {3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...} (ie. a discrete
> set of rationals).

Indeed, it was not Weierstrass. Weierstrass' method starts with multi-sets
(called aggregates) of non-negative rationals (i.e. sets where elements can
occur more than once), and is not what you state either.
A rational that can be exceeded by a finite sum of elements from such an
aggregate is a "component" of the aggregate. (So by definition, when a is
a component, all rationals smaller than a are components; and do not
confuse the term element with component.) There are diverging aggregates
(the set of components contains all non-negative rationals) and
convergent aggregates. Next he defines an equivalence relation on the
aggregates: two aggregates are equivalent if the sets of components are
the same. And the equivalence classes are the real numbers. So an
aggregate that is a representative for the real number pi could be:
[3, 0.1, 0.04, 0.001, 0.0005, 0.00009, ...], another one could be:
[1, 1, 1, 0.1, 0.01, 0.01, 0.01, 0.01, 0.001, ...]. (Note: order is
not important, multiplicity is.)

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). Next he defines the majorants of
such a set (a non-negative rational that is larger than each element)
and considered the set of majorants. The equivalence relation is that
two of the base sets are equivalent if the sets of majorants are equal.
And a real number is an equivalence class of base sets. It is more
cumbersome than what David Ullrich did, but keep in mind that set
theory was not yet well-established at that time, so there is not yet
talk about "set" of majorants. (Note also that Baudet has written that
what he did do was not different from what Dedekind did do, but was
easier.)

[ This all based on: "Het getalbegrip, in het bijzonder het onmeetbare
getal", Fred. Schuh, Noordhoff, Groningen, 1927.]
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

Quantcast