But in modern mathematics, the existence of infinite sets is simply assumed to be true—postulated as an axiom that does not require proof. Set theory is about more than describing sets.

Cantor’s set theory proved to be a powerful new mathematical system. But such abstract methods were controversial. “People were saying, if you’re giving arguments that don’t tell me how to calculate, ...

Ever since Cohen, set theorists have sought to shore up the foundations of infinite math by adding at least one new axiom to ZFC. This axiom should illuminate the structure of infinite sets, engender ...

So the first example of an infinite set. in mathematics is the set of all counting numbers. So one, two, three, four, five, six, seven, et cetera. That list goes on forever. That is an infinite ...

A set of the six sides of a die matches up one-to-one with a set of red shoes, helping illustrate Georg Cantor's set theorem. Image courtesy of GBH/NOVA It can be easy to take math for granted ...

For example, one infinity—the one most people are familiar with—is an infinite set of natural numbers: 1, 2, 3, and so on. However, there’s also an infinite set of real numbers, which ...

The colorable, divisible infinite sets in RT 2 2 are abstractions that have no analogue in the real world. And yet, Yokoyama and Patey’s proof shows that mathematicians are free to use this ...