```X-Message-Number: 21981
From: "michaelprice" < var s1 = "michaelprice"; var s2 = "ntlworld.com"; var s3 = s1 + "@" + s2; document.write("<a href='mailto:" + s3 + "'>" + s3 + "</a>"); >
References: < var s1 = "20030614090001.15566.qmail"; var s2 = "rho.pair.com"; var s3 = s1 + "@" + s2; document.write("<a href='mailto:" + s3 + "'>" + s3 + "</a>"); >
Subject: Transfinites
Date: Sun, 15 Jun 2003 05:56:35 +0100

Matthew S. Malek writes:

> As it turns out, there are actually different sizes of infinity,
> but only two of them.

Two?  There are infinite number of infinities (AKA transfinites)
To be more precise there are at least alepth_1 transfinites
where

alepth_0 = number of natural numbers

and

alepth_(n+1) = 2 ^^ alepth_n

and you can prove that

alepth_(n+1) > alepth_n.

i.e. 2 to the power of any transfinite is a larger
transfinite, which comes from the set theory result
that number of all possible subsets of a set is always
greater than the number of elements of the original set.
This allows us to construct a countable infinity
of transfinites.  We can then construct a greater
transfinite

beth_0 = sum (n =0 - infinity) alepth_n

which forms the basis of another hierarchy.  This
process of construction is never ending and generates
at least alepth_1 transfinites.

(Caveats:
I'm ignoring the continuum hypothesis or axiom,
which states that there may be more conjectural but
unprovable (in the Godelian sense) transfinites lurking
"in-between" the ones given by the above construction.
In which case the labelling scheme becomes more
complex.....
Also, I may be underestimating the number of
constructible transfinites; there might be alepth_2!)

> Which is the larger infinity:  The number of
> integers

alepth_0

>, or the number of [real] numbers between
> zero and one?

alepth_1

Cheers,
Michael C Price
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http://www.hedweb.com/manworld.htm

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