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Counting

Two sets can be put into a one-to-one correspondence if and only if they have exactly the same number of elements. For example:

{red,   yellow, green,    blue}
  |        |      |        |
{apple, banana, cucumber, plum}

You probably learned to count by putting things into a one-to-one correspondence with your fingers. Now you count by putting things into a one-to-one correspondence with a subset of the natural numbers (the numbers 1, 2, 3, ...). Like so:

{red,   yellow, green,    blue}
  |        |      |        |
{ 1,       2,     3,       4 }

In calculus you probably learned that "infinity" is not a number. They lied. Infinity, as a number, is represented by the symbol ₀ pronounced "aleph-null."

A set is denumerable if its elements can be put into a one-to-one correspondence with the natural numbers. Denumerable sets have ₀ elements.