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Math Trek

Small Infinity, Big Infinity

Infinity can be big or bigger, countable or not

5:30pm, January 8, 2008

Infinity is bigger than any number. But saying just how much bigger is not so simple. In fact, infinity comes in infinitely many different sizes—a fact discovered by Georg Cantor in the late 1800s.

Now a mathematician has come up with a new, different proof. Based on a simple game, the proof uses a strategy that might someday shed light on one of the great unsolved questions in mathematics.

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."

Lots of infinite things are countable. For example, suppose you take just the even numbers. They're countable,

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