Math CentralQuandaries & Queries


Question from Justin, a student:

Hello, I was just wondering why does the positive real infinity correspond to Aleph-null? Thanks a lot for answering my question!

All the Best,


It doesn't correspond very closely, but they do have in common the property that they can each be approached through a sequence of finite values. Larger transfinites cannot.

It is probably more useful to think of each of them being related in a different way to "omega", the first transfinite ordinal number. Aleph-zero counts the elements of omega if we "scramble" them, forgetting the order structure; and omega represents the order structure of a sequence such as (1,2,3...) that approaches positive real infinity.

Have you read that set-theory book yet? If you are having trouble finding one, you should be able to order Schaum's Outline of Set Theory for $20 or so through the publishers or any good bookstore. Dover Books
also has several good set theory texts for sale at excellent prices.

Good Hunting!

We received this note from Ami

Just wanted to share a few thoughts on Justin's aleph-null question.

First thing is that it's a bit unclear what Justin means by the "positive real infinity". I guess he means the set of real numbers greater than zero?

One thing that is important (though difficult) when thinking about infinity is to remember that when we talk about infinity in the context of alephs, we're actually talking about transfinite cardinals -- that is, "numbers" that describe the cardianlity, or 'size', of a set with infinitely many elements.

In this way, aleph-null corresponds to the 'size' (cardinality) of the smallest infinite set -- the set of natural numbers {1, 2, 3, ...}. Any set that can be put into a one-to-one correspondence with the set of natural numbers is considered to have the same 'size', cardinality aleph-null.

Now, back to Justin's question about the set of positive real numbers... There is a fairly sophisticated argument developed by Cantor which shows that the set of positive real numbers (for example) cannot be put into a one-to-one correspondence with the set of natural numbers -- in fact, the set of positive real numbers turns out to be BIGGER than the set of natural numbers (i.e. it has more elements), and so it's cardinality is *not* equal to aleph_null.

This raises the question: what is the cardinality of the set of positive real numbers? Unfortunately, there's no direct answer to that. The Continuum Hypothesis speculates that the set of real numbers is the 'next smallest' infinite set. However work by Godel and Cohen, two mathematicians of the 1900's, showed that
the Continuum Hypothesis could neither be proved nor disproved.... So the question remains open.

This is possibly not the answer Justin was looking for, but I hope it was helpful, or thought provoking, nonetheless :)


About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS