Some people don't want to say that there are just as many even natural numbers as natural numbers. But suppose that you and I will spend eternity singing numbers in harmony. You will sing every natural number in sequence: 1, 2, 3, ..., with a long pause for applause in between. And while you sing n, I will sing 2n. We will vary the speed of our singing to ensure that we take equal amounts of time. Clearly:
- The number of natural numbers = the number of your performances.
- The number of your performances = the number of my performances.
- The number of my performances = the number of even natural numbers.
- So, the number of natural numbers = the number of even natural numbers.