Re: Dedekind Cuts, Fundamental Sequences: why?

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 guess that stands
as a commentary on universities in the US.

Notice that the real numbers are not mentioned in that paragraph. Even
if you had not seen such a definition before, a moment's thought should
tell you that the definition of a lub in the context of the reals does
not actually use any property of the reals other than the fact that they
are (partially) ordered.

But it makes a whole lot of difference if the real numbers are assumed to
exists. I then have definitions for such expressions as x = sqrt(2), x <
sqrt(2) and x > sqrt(2), which I don't have if I am restricted to the
rational numbers.


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

David C. Ullrich
.

Relevant Pages

  • Re: Is one-to-one mapping valid for comparing infinite-sized sets?
    ... how it differs from rationals. ... For any non-empty set of reals ... but no LUB among the rationals. ... It looks to me as if the LUB (supremum, infimum, etc.) do not have any ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... That is not a definition of least upper bound. ... bound" the very first hit was the wikipedia entry for "supremum", ... LUB. ... properties of the reals other than the order. ...
    (sci.math)
  • Re: An uncountable countable set
    ... "counterexamples to standard real analysis" ... and whose lub is 0. ... I assume that if an upper bound is defined as: ... I don't see how this can happen for the reals though. ...
    (sci.math)
  • Re: Order Complete
    ... > an upper bound also has a least upper bound. ... any set of rationals whose upper ... bound in R is irrational has no LUB in Q. ... reals, it is trivial that every set of reals bounded above has a real ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... That is not a definition of least upper bound. ... bound" the very first hit was the wikipedia entry for "supremum", ... LUB. ... not actually use any property of the reals other than the fact that they ...
    (sci.math)