By Julie J. Rehmeyer
This is an excerpt from a post on MathTrek.
The smallest infinity is the one you'd get to if you could count forever. The numbers 1, 2, 3, 4 are called the natural numbers, and they are the most obvious example of this size of infinity. In honor of them, anything that has this size of infinity is called 'countable.'
Not everything infinite is countable, though. Take, for example, all the real numbers—all the counting numbers plus all the fractions plus all the irrational numbers. A real number is any number that can be expressed in decimals, though the expression might continue forever. For example, pi is a real number: It can be written as 3.14159.
What Cantor showed was that no matter what clever counting scheme you come up with, you'll never manage to count every last real number. He did it with a challenge: Just try it. Come up with a counting scheme, any counting scheme, and he'll find a real number you missed.