Re: Dedekind Cuts, Fundamental Sequences: why?

Dave Seaman wrote:

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.

The proof for convergence doesn't seem to require that distance be more than
rational if the limit is rational. Let {a_n} be a fundamental sequence of
rational numbers, i \in \N such that for all m,n>i =>
abs(a_m-a_n)<epsilon_i, where epsilon_i \in \Q. If for some q \in \Q and
for all Epsilon \in \Q and Epsilon > 0 we can find some j \in \N such that
abs(a_n-q)<Epsilon for all n>j, then we can say that {a_n} converges to the
limit q. We don't need real numbers for that, do we?

Therefore, there are
no metric spaces and no Cauchy sequences until *after* we have finished
constructing the reals, if we are to avoid circularity.

Stoll[*] introduces Cauchy sequences of rational numbers without first
defining the real numbers. I haven't read the entire development, but it
doesn't appear that he talks about limits until he defines the real
numbers. I don't believe he ever uses the term "convergence".

If I read III §1.2.2 (pp 2-4) correctly, the field of rational numbers
satisfy the conditions of a Hausdorff space, and therefore allows a series
to converge to a unique limit.

http://baldur.globalsymmetry.com/open-source/org/sth/math/behnke-et-al/vol-3.djvu

[*]Set Theory and Logic, by Robert R. Stoll
http://store.doverpublications.com/0486638294.html
(Yes, the
rationals are a metric space, but only *after* we construct the reals.)

As I mentioned to Virgil, Pickert and Görke attempted to make their
development self-contained with as little reliance on results from external
theories as possible. That is my reason for not wanting to admit metric
spaces into the development. I don't believe it is logically necessary.
It may be a satisfying and insightful way of proceeding, but I don't
believe it is the only way.

I may have been mistaken in what I said to Virgil regarding the point at
which they actually used the Cauchy criterion in connection with
convergence. That is, in proving the properties 61_1, 61_2 and 61_3 (page
141) which seem to follow from the definition of real number addition and
multiplication ((59) on page 139).

http://baldur.globalsymmetry.com/open-source/org/sth/math/behnke-et-al/vol-1.djvu

I wish they had simply provided the proof, or at least given an explicit
reference to where the proof, in the form they intend, can be found. I'm
pretty sure I can prove it using the Cauchy criterion without requiring any
assumption not yet stated. It seems far simpler to use (62) on page 141
with u=0. Am I missing something?

Every time we turn on our computers, they do something called booting.
Booting is shortened from "bootstrapping" which is a euphemism meaning to
lift oneself by one's own bootstraps. A similar situation arrises when we
get the latest source code for gcc (the gnu compiler collection). In order
to get a compiler from the source code we have to compile the compiler.
The only way to compile the compiler is with a compiler. These are all
examples of a self-referential system. The ultimate example of the
self-referential problem in Russell's Paradox. I am quite confident that
such a self-reference is at the core of the difficulty in defining the real
numbers in terms of the rational numbers.

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