Re: Dedekind Cuts, Fundamental Sequences: why?

David C. Ullrich wrote:

On Mon, 04 Jun 2007 07:53:56 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:

Dave Seaman wrote:

On Mon, 04 Jun 2007 06:32:31 -0400, Hatto von Aquitanien wrote:
David C. Ullrich wrote:

Every definition I have consulted for supremum and infimum begins
with the
real numbers. So to tell me that the reason we need to extent the
rational numbers to the real numbers is so that the domain of
numbers has suprema and infima assumes the real numbers to be
defined already.

Huh?

Definition: The ordered field F is complete if every nonempty
subset of F which is bounded above has a least upper bound.

That is not a definition of least upper bound.

I didn't say it was! It's a definition of completeness. You
know the definition of least upper bound, or I thought
you did.

"Every definition I have consulted for supremum and infimum begins with
the real numbers."

You couldn't have looked very far. When I googled for "least upper
bound" the very first hit was the wikipedia entry for "supremum", which
begins:

In mathematics, given a subset S of a partially ordered set T,
the supremum of S, if it exists, is the least element of T that
is greater than or equal to each element of S. Consequently, the
supremum is also referred to as the least upper bound, lub or
LUB. If the supremum exists, it may or may not belong to S. If
the supremum exists, it is unique.

Stupid me, I just looked in textbooks which I know to be in common use at
universities in the US for courses in real analysis.

You didn't look very far. Those textbooks are full of theorems that
depend on completeness.

I did not look for definitions of completeness. That's why I didn't say
that I had looked for such a definition.


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