Why does every power set have more elements than its original set?

The assertion that every power set has more elements than its original set is known as Cantor’s theorem. This theorem establishes that the cardinality, or numerical size, of any set is strictly less than the cardinality of its power set (the collection of all its subsets).1

For a finite set with $n$ elements, the power set contains $2^n$ elements, making the increase in size mathematically clear. For infinite sets, the proof is more complex and relies on Georg Cantor’s diagonalization argument. Cantor demonstrated that it is impossible to create a one-to-one correspondence between a set and its power set, as there will always be elements in the power set that cannot be paired with an element in the original set.12

This fundamental principle leads to an infinite hierarchy of transfinite numbers, proving that there are different "sizes" of infinity and that one can always construct a larger infinite set by taking the power set of a previous one.12