Re: Continuous injection from a subset of R^n to R

In article <456448F5.6000804@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 11/20/2006 10:46 AM, Virgil wrote:

Eckard Blumschein wrote:
David Marcus wrote:
Eckard Blumschein wrote:
Virgil wrote:

The real numbers will do.

Really? So far any attempt to furnish a well-ordering of the reals
failed. Already Aristotele understood why it is doomed to fail.

What did Aristotle say on the topic?

To every denominator there is a larger one.

Where did Aristoteles say that, and how is it relevant?

Relevant is the fact that the denominator can be enlarged indefinitely.

Rationals, when expressed as ratios of integers, have denominators, and
the denominator of any rational in this form can indeed be enlarged
indefinitely, provided the numerator keeps pace, but what relevance that
may have in well ordering anything is not at all apparent.

Here I have to admit a mistake of mine. Of course, I meant Archimedes,
not Aristoteles.
.

Relevant Pages

  • Re: Continuous injection from a subset of R^n to R
    ... So far any attempt to furnish a well-ordering of the reals ... Already Aristotele understood why it is doomed to fail. ... Where did Aristoteles say that, ... Relevant is the fact that the denominator can be enlarged indefinitely. ...
    (sci.math)
  • Re: Continuous injection from a subset of R^n to R
    ... Eckard Blumschein wrote: ... There is no standard well ordering of the rationals needed for any ... I doubt the issue of well ordering the reals ever presented itself to ... Aristoteles. ...
    (sci.math)
  • Re: Continuous injection from a subset of R^n to R
    ... Eckard Blumschein wrote: ... I doubt the issue of well ordering the reals ever presented itself to ... Aristoteles. ... The rest is simple reasoning. ...
    (sci.math)
  • Re: Continuous injection from a subset of R^n to R
    ... It is hardly a void promise when it has been so often validated by ... Can you point to a single case of obviously well ordered reals? ... Aristoteles. ... There is no smallest ratio, ...
    (sci.math)
  • Re: Continuous injection from a subset of R^n to R
    ... I doubt the issue of well ordering the reals ever presented itself to ... Aristoteles. ... I'll bite: How does Aristotle go from there being no largest number ... The rest is simple reasoning. ...
    (sci.math)