Re: Non-zero gaps between real numbers

On 30 nov, 11:45, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:
On Thu, 29 Nov 2007 07:33:59 -0800 (PST), Venkat Reddy





<vred...@xxxxxxxxx> wrote:
On Nov 29, 8:17 pm, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <56f03fcc-2b23-41f4-bb98-7eca6016b...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> Venkat Reddy <vred...@xxxxxxxxx> writes:
...
> > <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.
...
> But then here is the counter argument with an equal amount of validity
> - since there are no two different real numbers with zero gap between
> them, there is always a non-zero extent between any two real numbers,
> and hence real numbers can't fill the real line.

> According to you, is this a complete proof?

Apart from the "hence" part, that is also right: between any two real
numbers there is a non-zero extent. And we have also: between any two
real numbers there is another real number. But the "hence" part is
unclear. What does it *mean* that real numbers fill or do not fill
the real line?

Lets say that we can accept that the line is "filled" with some kind
of pieces, only when the gap between any two adjacent pieces is shown
to be zero. There is no reason why we should abandon this reasoning.

Why would we accept that? The line is in fact filled by real numbers,
but there's no such thing as two adjacent real numbers.

Why not just say "Let's accept that I'm right about everything"?

However, since there are no adjacent points there is no way to to show
that the gap is zero. Since there is no way to prove this, we can't
accept that line is filled.

- venkat

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

David C. Ullrich- Ocultar texto de la cita -

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If the geometric line is made of pieces at all, then it is made of
spatially adjacent pieces; this is obvious.

Reals/points are never adjacent.

Ergo, they are not the pieces the geometric line is made of.

If we contend that the analysis actually offers a numerical structure
isomorphic to a geometrical one, if we say that a line is (up to
isomorphism) a set of points, then we go too far.

The analysis treats the geometric continuum as a metrical space, and
for this purpose it only needs positions (points, reals) and distances
between points (differences between reals).

But why should we claim this exhausts the nature of the spatial/
temporal continuum?

It only reproduces it with absolute accuracy for purposes of numerical
computation.

The spatial/temporal continuum and the set of the reals are so
incomparable structures that the analysis can only get from the former
what goes through the sieve of the latter, what conforms to the holes
in the sieve.

Those holes are point-like.

Regards

.

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