Re: Dedekind Cuts, Fundamental Sequences: why?

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.

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.



--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
.

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)
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