How does Cantor's theorem compare a set with its power set?

Cantor’s theorem establishes that the cardinality (numerical size) of any set is strictly less than the cardinality of its power set—the collection of all its possible subsets.12

For a finite set $S$ with $n$ elements, the power set contains $2^n$ elements, making the relationship clear. However, Georg Cantor demonstrated that this principle also holds for infinite sets. Using a version of his diagonalization argument, he proved that it is impossible to establish a one-to-one correspondence between a set and its power set, meaning the power set is always strictly larger.12

This theorem is fundamental to set theory because it implies an unending hierarchy of transfinite numbers, as one can continue to take the power set of a power set to reach ever-increasing levels of infinity.12