From: Lee Corbin (lcorbin@tsoft.com)
Date: Mon Oct 14 2002 - 02:03:03 MDT
Ross writes
> [Spike wrote]
> > The really cool thing is that the integral you
> > describe demonstrates how an infinite number of
> > zeros can add up to a finite number. spike
>
> It uses a form of Lebesgue integral, it might use the F-Sigma Lebesgue
> integral, to determine a value for functions that sum to f(x)=1.
>
> I think that a sum of an infinite number of zeroes is equal to zero.
> The sum of an infinite number of infinitesimals may be zero
> (infinitesimal), finite, or infinite.
I believe that it depends on the cardinality of the number of
zeros. For example, we know that the area of the unit square
is one. But it consists of an uncountable number of points,
each of measure zero. Also, merely "infinitely" many, as
in countably many, are still zero. One must go from aleph 0
type infinity to aleph 1 type infinity to get a non-zero
result.
> It's kind of intuitive, people have a general idea of what a real line,
> it's a continuous path, the idea is to figure out what it means for
> anything to traverse the path. Luckily there is time, a time-like
> vector, to consider the paths in space, measured in space-like vectors.
It becomes an interesting difference between platonists and
formalists (or nominalists). We platonists *believe* in the
real line, and the real numbers, because we've used them a
lot, and they have readily identifiable properties, just like
the World Trade Center had readily identifiable properties,
and so was a real thing. (Yes, unicorns have well-known
properties too, but we (a) consider them not to exist in
the evolutionary record and (b) be too complicated to
exist in the same definite way that the number 17 exists.)
> Also it's consideration of how some numbers are rational numbers,
> they're fractions of integers, a/b for finite integers a/b, and the
> other real numbers are thus irrational. Irrational numbers have been
> proven to exist, for example e and Pi. My opinion is that rationals and
> irrationals alternate on the real number line, although some
> considerations of that are counterintuitive.
Pi and e are even "worse" than irrational. The square root of 2
is also irrational, but satisfies a polynomial equation, namely
x^2 = 2. But e and pi were proved to be *transcendental*, which
is to say that they do not satisfy any polynomial equation.
I don't think that viewing rationals and irrationals as alternating
on the real line is supportable. In the first place, as you know,
given a rational number, say, there is no *next* number, rational
or irrational either one. In the second place, the rationals
seem to be rather badly outnumbered: there are only countably
many of them, whereas there are uncountably many irrationals.
This superabundance of the irrationals is very striking in certain
examples, and gives one the feeling that rationals are very rare
creatures indeed.
> Heh, Bertrand Russell could prove Serafino to be the Pope.
Well, not anymore, because Russell has died (unfrozen).
It would be far easier for Serafino to prove that
Russell was once Pope. Being unable to prove stuff
is yet one more reason to avoid dying.
Lee
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