X-Message-Number: 21973
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 

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 

Again, the only importance of this stuff for most of us is the reminder that 
language is tricky and tradition is often badly mistaken.

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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