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From:  var s1 = "Ettinger"; var s2 = "aol.com"; var s3 = s1 + "@" + s2; document.write("<a href='mailto:" + s3 + "'>" + s3 + "</a>");
Date: Sat, 14 Jun 2003 11:36:20 EDT
Subject: infinities

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I had said that, by Cantor's method of counting, there are the same number of
odd integers as integers, but by another method there are twice as many
integers.

Matthew Malek writes in part:

> If there are infinite odd integers, how can there be twice as many even
> integers?
>
> "Twice infinity" doesn't really strike me as being a well defined term.
>
> As it turns out, there are actually different sizes of infinity, but only
> two of them.  Which is the larger infinity:  The number of integers, or
> the number of numbers between zero and one?

(First, I didn't say twice as many even integers--I said twice as many
integers as odd integers--although the other statement could also be proven by
appropriate definition.)

No, there are not only two (theoretical) infinities, but an infinite number
of them, cardinality aleph-null (the denumerable infinities such as the

integers), aleph-one, aleph-two, ad infinitum. The cardinality of the continuum,

called "c," may or may not be the same as aleph-one--this proposition is thought
to be undecidable.

How can there be twice as many integers as odd integers? By choosing your
definition of "twice as many."  Suppose I choose to say, "Set I of cardinality
aleph-null (integers) shall be deemed twice as large as set O of cardinality

aleph-null (odd integers) if O is a subset of I and a one-to-one correspondence
can be shown between the numbers n in O and the pairs of numbers (n, n + 1) in
I.

Again, the only importance of this stuff for most of us is the reminder that

Of course we remember too that the "infinities" in math are not actual or

completed infinities but only potential infinities in the sense of there always
being more. Whether there are any physical infinities in the universe, or
whether the universe itself is finite, are open questions.

Robert Ettinger

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