Infinities [was Re: The Big Bang]
Kennita Watson (kwatson@netcom.com)
Tue, 04 Nov 1997 22:55:42 +0000
John K Clark wrote:
> 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.
By this do you mean that there are B integers and 2^B points on a line, and
we don't know if there's anything in between?
Also, where do aleph-null, aleph-one, etc. fit into all this? My dictionary
says that aleph-null is "the first transfinite number". Is that what you call
A? If so, is aleph-one what you call 2^A?
Seeking notational unity and rudimentary mathematical understanding,
Kennita