Re: Non-zero gaps between real numbers
- From: William Elliot <marsh@xxxxxxxxxxxxxxxxxx>
- Date: Wed, 28 Nov 2007 23:14:39 -0800
On Wed, 28 Nov 2007, Venkat Reddy 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.
Read about Conway's surreal numbers. You'll like them.
They are a continumum of transfinite cardinal numbers (with roots even),
real numbers and infinitesimals. The infinitesimals pack the surreal line
with greater density than the reals. As the infintesimals get infinitely
smaller, the surreal continuum, in it's finest view, has scads of what he
call gaps. According to ZFC or what ever, these gaps may or may not
exist. Are they a gap in reality?
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
- References:
- Non-zero gaps between real numbers
- From: Venkat Reddy
- Non-zero gaps between real numbers
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