Re: Infinities

Hal Finney (hal@rain.org)
Thu, 6 Nov 1997 22:58:55 -0800

John Clark writes:
> On Thu, 6 Nov 1997 Hal Finney <hal@rain.org> Wrote:
> >This is not quite right; it is easy to show that 2^B equals C, where
> >B is aleph-null.
>
> It could be true, but it won't be easy to prove it.
>
> >Simply identify the points on the unit segment with their
> >representation in base two.
>
> But can you identify ALL the points or just some of them ?

This seems to be the main point. You can identify all of the points.
Every real number in [0,1] corresponds to an infinite binary fraction.
(The slight complication is that some numbers have two representations,
such as 1.0 and 0.1111..., the analog of 0.9999... in decimal. But this
can be dealt with since there are only countably many such numbers.)

Again, all that is needed is to agree that there is a one to one
correspondence between the real numbers and the infinite binary fractions
(modulo the point above). This applies to all the reals, even the
ones which are transcendental, random, or what have you. Each of them
corresponds to an infinite binary fraction.

Once you agree on this, it immediately follows that C = 2^aleph-null.

Hal