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 for any purpose, without permission, attribution, or notification."--LDC