Re: Axioms for positive elements of an ordered field?

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. ???

> 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.]


************************

David C. Ullrich
.

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