Re: counter example in analysis

Eckard Blumschein wrote:
On 11/6/2006 9:15 PM, Dave L. Renfro wrote:
Eckard Blumschein wrote:
Nonetheless, according to Dedekind's wording, any
countable amount of rational numbers is overall dense.

No, and he never said this either. Perhaps you could
give a precise quote and reference?

I prefer Fraenkel 1923 because this book has a register: The original
wording was on p. 106 "ueberall dicht". Admittedly, my wording "any
countable amount of rational numbers" was incorrect. I meant: The
unlimited, potentially infinite amount of rational numbers it is thought
to be overall dense, and not even this corresponds to the original. Why
did I not follow to Cantors notion of "lineare Punktmengen" (linear sets
of points)? The given Defintion was: "A set of points is considered
overall dense if there is always one more point between two selected
points." I looked at how the points were described: "Eine genügend
starke Besetzung der Linie mit Punkten" [A sufficiently strong trimmings
of the line with points] as well as the metaphor of an amplifying
microscope clearly indicated potential infinity. The genuine continuum
cannot be resolved into points no matter how many times one zooms. So
the definition does not really refer to actual infinity, at least not
immediately.
My "any countable amount" should read "a potentially infinite amount".

Oh, well, that clears that up.

You know, words have precise meanings in mathematics.

--
David Marcus
.

Relevant Pages

  • Re: counter example in analysis
    ... countable amount of rational numbers is overall dense. ... potentially infinite amount of rational numbers it is thought ... to be overall dense, and not even this corresponds to the original. ... of rationals, nor any other densely ordered set. ...
    (sci.math)
  • Re: counter example in analysis
    ... countable amount of rational numbers is overall dense. ... to be overall dense, and not even this corresponds to the original. ... microscope clearly indicated potential infinity. ... of rationals, nor any other densely ordered set. ...
    (sci.math)
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