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