Re: Non-zero gaps between real numbers

On Nov 29, 6:34 am, Venkat Reddy <vred...@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.
Because a complete proof would require one to find two different real
numbers with zero gap between them. 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

I'm not sure I understand you. Is the following what you mean?

1. Let's locate all the reals on the line as points.
2. There will be no contiguous reals/points.
3. But, by choosing the suitable reals/points, the distance can be
rendered less than any given positive numerical quantity (but not
zero). This brings a limit into scene.
4. If we compute that limit as zero, the points become ultimately
contiguous and the line collapses into the zero dimensional.
5. So, we must compute the limit as an infinitesimal extent.
6. That infinitesimal quantity measures...what?

The minimal distance between points? It happens that there is no
minimal distance between points in the model. All distances are of
positive real extent and for any distance there is a smaller one.

I agree that a line cannot be thought of as consisting of points.
Points represent positions in the line, they are not the stuff the
line is made of. So, the reals and the line are structures of just
different nature, inconmensurable, so to say.

But I don't see this implies the existence of infinitesimal gaps
between the points.

Regards









.

Relevant Pages

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