How does the power-set axiom create a set of all subsets?

In axiomatic set theory, such as the Zermelo-Fraenkel system (ZFC), the axiom of power set is a fundamental principle used to construct new sets from existing ones.1

The axiom asserts that for any given set $A$, there exists a set that contains all the subsets of $A$ as its members. This resulting collection is known as the power set, symbolized as $P(A)$. By defining the power set as a legitimate set within the theory, the axiom allows mathematicians to move beyond the limitations of naive set theory—where unrestricted collection often leads to paradoxes—and instead build an iterative hierarchy of sets.213

This construction is essential for developing the theory of transfinite cardinal numbers. According to Cantor’s theorem, the power set of any set $A$ always has a strictly greater cardinality than $A$ itself ($|A|<|P(A)|$). By repeatedly applying the power set operation, one can generate an unending hierarchy of increasingly larger infinite cardinalities.2