Re: Dedekind Cuts, Fundamental Sequences: why?

James Burns wrote:

Hatto von Aquitanien wrote:
David C. Ullrich wrote:
On Mon, 04 Jun 2007 05:37:27 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:
David C. Ullrich wrote:

The fact that the reals are complete is used in many
places - analysis simply would not work without it. You're
just at the start...

That begs the question.

Yes it does. When you know some analysis it will be very clear
to you why completeness is so important.


That still begs the question. The fact that something _is_
useful does not explain _why_ it is useful.


Please excuse me for butting in here, Dr. Ullrich.

------------

Whether it begs the question depends on which question
is on the table. Judging from the subject header, your first
question seems to have been "Why are Dedekind cuts used to define
reals?" (And partitions of Cauchy sequences as well?)
That question seems to have been answered well, by several
people, not just Dr Ullrich.

<quote>
What is the step of logic which leads one to seek an extention of the
rational numbers to the real numbers?

I understand the arguments given by Prickert and Görke for all of the
previous extentions.  But when they go from the rationals to the reals,
they don't really present a formal equation in need of a solution as they
had for all of the prior extentions.  It leaves me to wonder what, exactly,
the criteria for success is.
</quote>

I still don't believe I have a good answer to my question. People presented
examples such as the solution to x^2=2 as an equation in need of solution.
As I have already pointed out, that is a single example, and not a general
form. This is different from all of the previous steps in the development
where there was a clearly stated formula which was in need of solutions not
provided by the domain of numbers already developed. I'm presented with
three ways of defining the real numbers. One is to use infinite decimal
representation. The second is Dedekind cuts, and the third is fundamental
sequences.

I am assured by the authors that proofs using the first definition are
prohibitively lengthy. That leaves them to expound on Dedekind cuts and
fundamental sequences. I'm told that ensuring every set of rational
numbers has a supremum will "plug all the holes" in the rational numbers.
I don't see how that is different from saying "if there's a hole, fill it
with a real number". Why does {x:x \in \Q and x<sqrt(2)} not have a
supremum in \Q? "There's a hole at sqrt(2)."

As I have already stated, the motivation for either of these two approaches
appears to be so that the algebraic features of the field of rational
numbers can be carried over to the real numbers. Weyl flat out rejected
the approach using terms such as "nonsense", "completely wrong", "not even
a shadow of a proof" and "blatant circulus vitiosus".


Because it is too, too convenient to give up saying
these two algorithms calculate the same number and
imagining that number as a dimensionless, structureless
point on the real number line, we do say that they
calculate "the same number". What is this number they
calculate? In my view, that dimensionless, structureless
point is "really" the set of all the good algorithms
that calculate its position, which is to say, the equivalence
class of Cauchy sequences that converge to that same point.

And what is that point again? The Cauchy sequences
that converge to it. It all seems very circular, but
built up, step by step, from rationals, to sequences
of rationals, to Cauchy sequences, to partitions of
Cauchy sequences, it's all well-founded and consistent.

Not too long ago I proposed a possible means of defining the real numbers as
a set of algorithms which are left in symbolic form until there is a need
to arrive at numerical results. I didn't pursue it very vigorously because
it seems like more work than it's worth.

I don't particularly like the way Weyl formulated vector spaces in what has
now become received dogma. I can only interpret his reasoning to be that
he rejected the extention of rational number multiplication to the real
numbers, in general. There is nothing in the original concept of a finite
dimensional vector space formed of n-tuples that would preclude an
extention from rational to real component addition thence to multiplication
if such were deemed possible for "scalars".

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