Re: Dedekind Cuts, Fundamental Sequences: why?

On Wed, 06 Jun 2007 00:08:20 -0400, Hatto von Aquitanien wrote:
Dave Seaman wrote:

On Tue, 05 Jun 2007 09:57:58 -0400, Hatto von Aquitanien wrote:
Aatu Koskensilta wrote:

On 2007-06-05, in sci.math, Hatto von Aquitanien wrote:
That tells me how they are useful, not why. But what I really meant by
my previous comment is that he was begging the question as to why
fundamental
sequences or Dedekind cuts are useful in defining completeness. And I
still do not know if there was a clearly defined objective which can
subsequently be verified as being accomplished.

They provide a concrete way of specifying a complete ordered field with
a countable dense subset, reducing talk about reals to talk about
naturals and sets of naturals.

That is where Weyl rejected the development. He basically argues that
using sets of rational numbers as a means of defining the real numbers is
not
supported by any form of proof that it is even meaningful. It is also
fairly clear that he doubts such a proof could be given.

I am certainly no expert on Weyl, but your description implies that he
considered a "meaningful proof" to be something more than merely a
*valid* proof. What is the difference, exactly?

I guess this is evidence that I should refrain from posting after studying
for more than 20 hours without a break. I meant to write "supported by any
form of proof, _or_ that it is even meaningful"

We don't prove definitions. We prove theorems. Either the Dedekind cuts
or the Cauchy sequences provide a definition, and the number system so
defined is provably a complete ordered field. The concept of a complete
ordered field is certainly meaningful in analysis.

It is not completely clear to me that these definitions do reduce all
discussion of real numbers to discussions of rational numbers. In places
where I can perceive discussions of real numbers being derived strictly
from concepts related to rational numbers, I don't see either fundamental
sequences nor Dedekind cuts add anything essential, or even useful.

I don't follow this. Which of the following are you claiming?

(1) The constructions mentioned do not succeed in producing a complete
ordered field, or
(2) The notion of completeness is not essential or useful in
mathematics?

Neither one. I was speaking "places where I can perceive discussions of
real numbers being derived strictly from concepts related to rational
numbers".

The fact that the reals are a complete ordered field can be derived
strictly from concepts related to rational numbers. What other
properties of the real numbers do you have in mind?

Such a reduction is of importance in that, provided
we pretty much agree what sort of principles and modes of reasoning are
acceptable when dealing with naturals and sets thereof, we get a clearer
view of what sort of reasoning about reals, functions of reals, and so
on, is acceptable.

That is another aspect of Weyl's argument. He claims the reasoning used
to extend to rational numbers to the real numbers is cyclical.

Cyclical? Do you mean circular?

Yes. I guess I translated the Latin incorrectly due to fatigue.

How so?

"It is probably not necessary to repeat that it would be meaningless to
include among these principles an assertion such as the following: If A is
a property of properties, then one may form that property P_A which belongs
to an object x if and only if there is a property constructed by means of
these principles and itself possesses the property A. That would be
blatant circulus vitiosis; yet our current version of analysis commits this
error and I consider this ground for censure."

Can you be more specific? Exactly where do you see circularity in the
construction of ther real numbers from the rationals?

For myself,
if I do allow for some arguments founded in terms of the rational numbers
to indicate that the rational numbers are in need of extention, it is not
clear to me that all such circumstances are, or can be addressed in this
way. For example, I can give an algebraic argument for wanting a
solution
to 2=x^2. The motivation, at least, arrises from within the context of
the
rational numbers. Pi, OTOH, seems to be a bird of a different feather.

The motivation for completing the rationals lies mainly in analysis, not
algebra.

My point was that using the algebra of the existing domain I can construct
expressional forms which sometimes, but not always produce a number within
the existing domain. That was the line of development which led to each
previous extension.

Consider the mean value theorem, for example.

That hasn't been introduced yet. It's covered in volume II.

Its proof
depends heavily on the completeness of the reals, and this theorem lies
behind much of analysis.

Why not just extent the natural numbers immediately to the positive real
numbers using infinite decimal expression, introduce 0 and negation and be
done with it?

Because it is clumsy and inelegant, and because the integers and the
rationals are worthy of study in their own right.

I am assured that development of the algebra of real numbers is excessively
unwieldy when approached in this way, but I have not seen that
demonstrated.

The definition of addition is quite unwieldy. There is no upper bound on
the length of a carry that might occur. Multiplication is even worse.

There are constructs which we perceive to warrant inclusion in the set of
things we call numbers. IOW, when we view the field of rational numbers,
and compare it with our expectations of completeness, it comes up short.
We can identify constructs which lie outside the set which we want to
treat
as numbers. Fundamental sequences and Dedekind cuts appear to me to be
simply an elaborate loophole set up to allow the inclusion of everything
our intuition tells us aught to be in the set of numbers (excluding
Gaussian numbers, which is another can of worms, or the same can from the
other end.)

Is the construction of the rationals from the integers also an enormous
loophole, in your mind? Please explain.

No. Each of those steps was taken after explictly and clearly stating the
objective.

The objective in this case is to achieve completeness. In fact, the
process involving Cauchy sequences is called "completion of the
rationals". Even if you don't fully understand the reason that something
was done, I don't see how that makes it a "loophole".


--
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: Dedekind Cuts, Fundamental Sequences: why?
    ... sequences or Dedekind cuts are useful in defining completeness. ... The constructions mentioned do not succeed in producing a complete ... view of what sort of reasoning about reals, functions of reals, and so ... The motivation for completing the rationals lies mainly in analysis, ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... I don't know if there is a way of formally defining pi in terms of rational ... there are uncountably many reals and only ... using the rationals as a starting point. ... In cases where the Cauchy sequence (or Cantor sequence obeying the Cauchy ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... sequences or Dedekind cuts are useful in defining completeness. ... sequences nor Dedekind cuts add anything essential, ... view of what sort of reasoning about reals, functions of reals, and so on, ... The motivation for completing the rationals lies mainly in analysis, ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... if the only applications of completeness are things ... What you should do is _perform_ the construction of the reals ... Is the construction of the rationals from the integers also an enormous ... not complete - we want to extend them to a complete ordered field, ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... there are uncountably many reals and only ... using the rationals as a starting point. ... mean specifically defining every member of the set. ... Once we have finished constructing the reals, we have a perfectly good order ...
    (sci.math)
Loading