# Re: Infinities [was Re: The Big Bang]

Lee Daniel Crocker (lcrocker@mercury.colossus.net)
Wed, 5 Nov 1997 15:05:26 -0800 (PST)

> > 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

It is generally believed (I don't follow the mathematics culture
enough to know if it has been proven or not) that the "continuum"
(i.e., the number of points on a line or plane or space) is indeed
Cantor's Aleph-1. Aleph-0 is the cardinality of the set of
integers; Aleph-1 is the number of possible subsets of integers,
or 2 ^ Aleph-0. Aleph-2 is the number of possible subsets of a
set of cardinality Aleph-1. If Aleph-1 is indeed equivalent to
the continuum, then Aleph-2 would be the cardinality of the set of
all possible drawings, for example (where "drawing" is a subset of
points on a plane). I can't think of a useful example of a set of
cardinality Aleph-3 offhand.

It /is/ provable that the continuum /must/ be greater than Aleph-0
(this was Cantor's ingenious "diagonal" proof).

```--
Lee Daniel Crocker <lee@piclab.com> <http://www.piclab.com/lcrocker.html>
"All inventions or works of authorship original to me, herein and past,
are placed irrevocably in the public domain, and may be used or modified