Foundations of Mathematics

Set Theory

What are some strange characteristics of infinity?

There are some strange ideas about infinity—the most interesting one being that it’s not just a bunch of numbers that go on forever. There are certain properties, too. For example, infinity doesn’t always equal infinity; and infinity minus infinity does not equal zero; or even infinity over infinity (infinity/infinity) does not equal one. Why don’t the “normal” rules of mathematics apply here? The biggest problem is infinity itself—or is it “themselves”? After all, one infinity in the questioned equation may be larger than the other—and you wouldn’t know.

There is also the paradox of infinity. For example, take the following two sequences:

ab = 1, 2, 3, 4, 5, 6 …

cd = 2, 4, 6, 8, 10…

ab contains all natural numbers, while cd contains all even natural numbers. Although you would think cd would have half the number of terms, and thus would be a “smaller” infinity, they both have an infinite number of terms— thus the paradox of infinity. That is because most of us think of infinity as a number rather than a concept.


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