Why is the power set of an infinite set larger than the original set?

The fact that the power set of any set—including an infinite set—is strictly larger than the original set is established by Cantor’s theorem.

In set theory, the size of a set is defined by its cardinality. For any set $A$, the power set $P(A)$ is the collection of all possible subsets of $A$. Cantor proved that there is no one-to-one correspondence between a set and its power set; specifically, the cardinality of $P(A)$ is always strictly greater than the cardinality of $A$.12

This proof relies on a version of Cantor’s diagonalization argument. By demonstrating that the power set operation always produces a larger infinity, Cantor revealed an unending hierarchy of transfinite numbers, such as ${\aleph }_0$ (aleph-null), ${\aleph }_1$, and beyond. This result is fundamental to modern mathematics, as it shows that there are different "sizes" of infinity and that one can always construct a larger infinite set by taking the power set of the previous one.123