## Sizes of Infinity

Everyone knows about infinity, but most people don’t know that there are actually different sizes of infinity: the transfinite numbers. This may out odd at first; after all, how do you define the size of something that is infinite?

For example: there are an infinite number of even numbers. But, to be more precise, there is a “countable infinity” of even numbers; in other words, you can enumerate them.

Say we start at 0 and call it the first even number; then we say that 2 is the second even number, -2 is the third, 4 is the fourth, -4 is the fifth, et cetera. In this way we can assign an integer to each even number. (Note that we can choose any numbering scheme we like; what’s important isn’t *how* we enumerate them, but rather *that* we can.) Therefore, the size of the even numbers is as big as the size of the integers, and that size is “countable infinity”, often represented by the symbol aleph zero. A similar argument can be given for odd numbers, positive numbers, negative numbers, primes, and so forth.

On the other hand: when one tries to count the real numbers, a problem arises. For any two real numbers there will always exist another between them. It is not possible to create a one-to-one correspondence between the real numbers and the integers; in other words, they are impossible to count. For a rigorous proof of this fact see Cantor’s diagonal argument.

Notice that between any two rational numbers (numbers of the form a/b) there is also always another number; nevertheless, we *can* construct a an isomorphism between the rationals and the integers. So, in fact, there are just as many rationals as there are integers! One such isomorphism is the Calkin-Wilf tree.

### Notes

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