the infinite(s)

Anton Sherwood (
Wed, 5 Nov 1997 19:22:33 -0800 (PST)

John Clark writes
: 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.

Isn't that the "Continuum Hypothesis", which has been proven
unprovable either way? So it's an arbitrary choice of axiom.

Kennita asks
: 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?

Yes, if B = aleph-null. Think of the set of real numbers (points
on a line) as expressed in binary: B bits running to the left, B bits
running to the right, thus the members of the set are 2^B. Cantor
used a charming trick to show that this set is non-countable, i.e.
cannot be mapped one-to-one with the set of integers.

Anton Sherwood *\\* +1 415 267 0685 *\\*