Re: Non-zero gaps between real numbers

On Wed, 28 Nov 2007 21:34:33 -0800 (PST), Venkat Reddy
<vreddyp@xxxxxxxxx> wrote:

The definition of real numbers allows one to find a real number
between any two given different real numbers. If one uses this to
assert that there is no non-zero extent gap devoid of real numbers in
it, then I think it is not a complete proof but just an assertion.

You're simply wrong - it _is_ a complete proof.

This sort of nonsense is why I find the the people who
are not morons but who nonetheless try to bend over
backwards to find reasonable interpretations of the
things you say so puzzling.

Because a complete proof would require one to find two different real
numbers with zero gap between them.

Uh, no, a complete proof that "there is no non-zero extent
devoid of real numbers" does not require that.

This is simply nonsense, like claiming that a complete proof
that there are no unicorns would require that we find an
elephant and then show that it is not a unicorn. You have
no idea what "proof" means in mathematics.

Since this is not possible, one
could also say that the gaps are never filled by real numbers.

Since none of these "half-proof"s is more valid than the other, I
don't see why we should dismiss one in the favor of the other.

Actually, by assuming that real numbers can fill the continuum, we are
entering into an unknown realm and forced to prove that zero extent
points can fill the finite interval in continuum.

On the other hand, there is no ambiguity in accepting non-zero gaps.
Since we could never find adjacent numbers with zero gap, there is no
harm in stopping at infinitesimal level and take it is the model of an
extent. This persistence of non-zero extent is safe to stick to, and
it reflects extent in a fractal sense, without collapsing to zero at
any level.

I think we have collapsed things to points because someone didn't like
the never-ending recursion seen in the forms such as fractals, and
wanted to nail down things to zero, which is sad.

I don't think we should purge infinitesimals which seem more sensible
than the argument of points making up the space.

- venkat



************************

David C. Ullrich
.

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