Re: Dedekind Cuts, Fundamental Sequences: why?

Glen Wheeler wrote:

"Hatto von Aquitanien" <abbot@xxxxxxxxxxxxxx> wrote in message
news:FqWdncA-GfEgP_HbnZ2dnUVZ_g-dnZ2d@xxxxxxxxxxxxxxxx
<restored>
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?

</restored>

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.


The only difficulty my friend is in your mind. There are many ways to
define the reals that do not presuppose the existence of the reals. See
for example (IIRC) Seaman's message and my reply for two consistent
methods.

Did I say that it had not been done?

I can go through all the arguments with sufficient understanding to satisfy
course requirements. That doesn't mean I have convinced myself of the
validity of every step. I have already pointed out that there is a problem
with the approach of using Dedekind cuts in that it relies on a theory of
sets which has not been shown to be free of contradictions. The same
source also asserts that similar problems arise in all approaches to
constructing the real numbers. But that really was not the point I was
addressing with my comments.

At this point in the thread, you are simply trolling. All of this is
covered (at least in Australia, I presume other countries are at least as
good if not better) in a second year, compulsory analysis course.

I was told that: "[W]e can'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."

Now, the development I am reading speaks of "fundamental sequences of
Cantor", and says that the term "Cauchy sequence" is synonymous. If there
is a difference between the "fundamental sequence" and "Cauchy sequence"
which does not involve spelling, pronunciation or historical reference, I
would like to know what the difference is. To say that fundamental
sequences don't converge because they exist in an incomplete space, seems
to be nothing more than semantic bookkeeping, if indeed that is even a
valid interpretation of the usage.

"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."

That is very much in agreement with the usage I am familiar with.

"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."

That does not appear to be a necessary restriction. It may be a useful way
of formulating the development, but according to my understanding of the
material I referenced and provided access to (at my own personal expense, I
will add,) it is possible to talk about limits of sequences in \Q. I can
also construct the argument from first principles to show that it is
meaningful to talk about a sequence approaching a limit in \Q, and that a
fundamental sequence may have a limit in \Q.

Notice that my comments are about mathematics, and not about Glen Wheeler.

--
http://www.dailymotion.com/video/x1ek5w_wtc7-the-smoking-gun-of-911-updated
http://911research.wtc7.net
http://vehme.blogspot.com
Virtus Tutissima Cassis
.

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