How does Cantor's theorem show that there are infinitely many different sizes of infinity?

Cantor’s theorem demonstrates that the cardinality (numerical size) of any set is strictly less than the cardinality of its power set—the collection of all its subsets.1

While this relationship is intuitive for finite sets, Georg Cantor proved it applies to infinite sets as well. By using a diagonalization argument, he showed that the power set of an infinite set is always more numerous than the original set. Because one can repeatedly take the power set of a set to generate a new, larger set, this process creates an infinite hierarchy of increasingly larger transfinite numbers.123

This discovery fundamentally changed mathematics by proving that infinity is not a single, uniform concept but exists in many different sizes. For instance, while the integers are countably infinite, the real numbers are uncountable and represent a larger magnitude of infinity.213