Greg Butler <gsb1997@ix.netcom.com> On Tue, 4 Nov 1997 Wrote:
>I don't understand how one can compare one infinite number with
>another
The rules of arithmetic are very different and non intuitive when you deal
with infinite numbers, mathematicians call them Cardinal Numbers. The rules
for infinite arithmetic were discovered by the German Mathematician
George Cantor (1845-1918). Today he is universally recognized as one of the
greatest mathematicians in the last 100 years, but during most of his
lifetime he was thought a fool by his colleagues and treated quite shabbily.
They thought the idea of mathematical infinity was stupid. He died penniless
and quite insane in a mental hospital. These ferocious attacks a century ago
on the leading advocate of mathematical infinity remind me a little of the
attacks on the leading advocate of physical infinity today. I hope Cantor's
fate does not befall to poor Tipler.
Cantor proved that the number of integers, call it A, is the smallest
cardinal number. He proved that the amount of even numbers is the same as
the amount of all numbers. He proved that N*A =A if N is any finite number,
and he even proved that A*A =A. However he also proved that 2^A is NOT
equal to A.
Cantor's crowning achievement was when he proved something we now call
Cantor's Theorem, it states that if B is any cardinal number then B < 2^B.
This means there are an infinite number of cardinal numbers, an infinite
number of different infinities. However he was not able to figure out if
there is an infinite number between the number of integers and the number of
points on a line, and even today it is not known.
>how big is an infinite amount of space squared? Is it bigger than a
>"regular" infinite amount?
No. The number of points in a line or in a square or in a cube or in a
hyper-cube of finite dimensions are the same, that is, you can put them in a
one to one correspondence.
John K Clark johnkc@well.com
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