From: Ross A. Finlayson (extropy@apexinternetsoftware.com)
Date: Mon Oct 14 2002 - 00:19:44 MDT
On Saturday, October 12, 2002, at 01:59 PM, scerir 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
(infinitestimal), finite, or infinite.
It's like considering a square and asking, is it just a square, or four
smaller squares? Drop a pencil on the square and call the mark a
point. It's like wondering if a line segment is a set of points, and
how.
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.
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.
Infinity and infinite sets are a part of several areas of mathematics,
including analysis and set theory.
> 0 = 0 + 0 + 0 + 0 + ...
> 0 = (1-1)+(1-1)+(1-1)+(1-1)+ ...
> 0 = 1-1+1-1+1-1+1-1+ ...
> 0 = 1 +(-1+1)+(-1+1)+(-1+1)+ ...
> 0 = 1 + 0 + 0 + 0 + ...
>
>
>
If you look at the sine function, it's average derivative is zero, it
ranges from -1 to 1, sin(x) in the limit as x diverges to infinity is
said not to exist. The sinc function, f(x)/x in the limit has at point
x a value between 1/x and -1/x, which as x diverges puts the value of
sinc(x) between 0+ and 0-, or that the limit of sinc(x) as x diverges is
zero. The limit of sinc(x) is not always said to exist.
Heh, Bertrand Russell could prove Serafino to be the Pope.
Ross
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