Re: Dedekind Cuts, Fundamental Sequences: why?

On Sat, 09 Jun 2007 01:34:05 -0400, Hatto von Aquitanien wrote:

"Cauchy's Criterion for Convergence. In accordance with Cantor's basic
idea, real numbers can be described by convergent rational sequences. Two
rational sequences (r_n) and (s_n) have the same (real) limit, if and only
if the sequence of their differences (r_n-s_n) converges to zero. It is
natural therefore to _define_ the _real_ _numbers_ as equivalence classes
of _convergent_ _rational_ _sequences_ ; two sequences being equivalent
when their difference sequence converges to zero. _For_ _this_
_definition_ to be meaningful, the _convergence_ of a sequence has to be
_characterized_ _without_ making _use_ of _limits_ . This can be done with
the help of Cauchy's criterion, which will be used to define the sequences
concerned."

http://www.amazon.com/Numbers-Graduate-Texts-Mathematics-Readings/dp/0387974970

Although the consensus in sci.math seems to be otherwise, the way I
actually learned it in my graduate real analysis course is that we con't
speak of "Cauchy sequences of rationals" when constructing the real
numbers. We speak of "fundamental sequences of rationals" instead.
I believe this is Cantor's terminology.

A sequence {a_n} of rationals is said to be "fundamental" if, for every
*rational* epsilon > 0, there exists N > 0 such that | a_m - a_n | <
epsilon for every m,n > N.

The reason for this is that the definition of a Cauchy sequence makes
sense only in a metric space, and the definition of a metric space
requires the metric to be a real-valued function. Therefore, there are
no metric spaces and no Cauchy sequences until *after* we have finished
constructing the reals, if we are to avoid circularity. (Yes, the
rationals are a metric space, but only *after* we construct the reals.)


--
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/>
.

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