Non-zero gaps between real numbers

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


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Relevant Pages

  • Re: Non-zero gaps between real numbers
    ... although the assertion would be easily provable by other means. ... numbers with zero gap between them. ... Uh, no, a complete proof that "there is no non-zero extent ...
    (sci.math)
  • Re: Non-zero gaps between real numbers
    ... assert that there is no non-zero extent gap devoid of real numbers in ... numbers with zero gap between them. ... Uh, no, a complete proof that "there is no non-zero extent ...
    (sci.math)
  • Re: Single and double data types
    ... What *does* happen in VBA is that sometimes it incorrectly returns the ... smallest storable number, and sometimes it rounds down to zero, but it ... Doing a calculation that results in an underflow will again give bogus ... >The result of the calculation could fall into the gap. ...
    (microsoft.public.word.vba.general)
  • Re: any() and all() on empty list?
    ... This is trivially true in the case of the empty list. ... def any: ... zero = any ... assert == one ...
    (comp.lang.python)
  • Re: Non-zero gaps between real numbers
    ... > - since there are no two different real numbers with zero gap between ... There is no reason why we should abandon this reasoning. ... Think of the spatial continuum as it appears to your spatial ...
    (sci.math)