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Ask Dr. Knowledge

What does it mean to have more than one kind of infinity? Isn’t infinity just infinity?

September 5, 2011

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What does it mean to have more than one kind of infinity? Isn’t infinity just infinity?

Infinity is a subtle concept.

The modern conception of infinity derives from thinking about sets - collections of objects. A simple set of objects, say six apples, is easy to measure. But some sets are more complicated.

Take the set of all counting numbers, 1, 2, 3, 4 . . . and so on. Mathematicians call such sets countably infinite.

Within the set of all counting numbers, there are other infinite sets - for example, the set of all even numbers and the set of all odd numbers. Both of these sets are infinite, and they are also subsets of another infinite set. All three are equally infinite, even though the first one contains the second two.

Weird, but that’s infinity.

Even weirder, imagine all the real numbers that exist between 0 and 1. They are infinite, but that infinity is not countable, because no matter how you try to list them you always find a way to make one number that’s not on your list.

So the integers are countably infinite, while the numbers between them are uncountably infinite - two kinds of infinity.

There is a lot more to be said about infinity but the column length is, unfortunately, finite.

Ask Dr. Knowledge is written by Northeastern University physicist John Swain. E-mail questions to drknowledge@globe.com or write to Dr. Knowledge, c/o The Boston Globe, PO Box 55819, Boston, MA 02205-5819.