You might be wondering why we care about 1-to-1 correspondence, since there is another way to compare the sizes of sets: just count the elements of each.

The answer is that sometimes we * can’t *count the elements of a set. But a 1-to-1 correspondence still works.

For example, if a set is infinite, we can’t first count all the elements in it and then compare the result to another set. The counting would never end!

Take the set of natural numbers, which we’ll symbolize with the capital letter N.

N = {0, 1, 2, 3, 4 …}

0 is a wonderful number. Don’t hate on 0!

You should know, sometimes 0 is not called a natural number. That terminological dispute isn’t important, however. We will include 0 in the set N.

What is the cardinality of N?

Well, there is no finite number that is the cardinality of N. The ellipsis “…” means N just keeps going without end.

|N| is an infinite number.

But we also said that every set has a size. So |N| is an infinite number.

We use the symbol ℵ_{0}, pronounced aleph-naught, for the cardinality of N. (Aleph is the first letter of the Hebrew alphabet.)

ℵ_{0} is an infinite number.

ℵ_{0} is an infinite number.

In grade school your teacher might have told you that “infinity is not a number.” Well, that’s false.

Infinity is not a * natural number*. Maybe that is what your teacher meant, and that is true. Every natural number is finite.

ℵ_{0} really is a number. It’s just an infinite number.

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