Re: Dedekind Cuts, Fundamental Sequences: why?

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?

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?

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? How so?

Are you aware that the rationals are defined in terms of the integers,
and the integers in terms of the natural numbers? Do you feel that those
constructions are circular, as well? Do they add anything essential or
useful to mathematics?

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. Consider the mean value theorem, for example. Its proof
depends heavily on the completeness of the reals, and this theorem lies
behind much of analysis.


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.



--
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?
    ... sequences or Dedekind cuts are useful in defining completeness. ... The fact that the reals are a complete ordered field can be derived ... rationals are worthy of study in their own right. ...
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  • Re: Dedekind Cuts, Fundamental Sequences: why?
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    (sci.math)
  • Re: Are the Dedekind cuts uncountable?
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  • 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, ...
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