Why are power sets called power sets?

Question

Why are power sets called power sets? What is so powerful about them?

Answer 1

When you have a set A with $|A|=N$, we have $|\wp(A)|=2^N$, 2 to the power of $N$.

That lead to the name power set.

The notation $2^N$ came from the fact that this was the case when $A$ is finite.

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In German, a Potenzfunktion is a fuction in the form of $X^n$. The name Potenzmenge (power set) appears to come form Untersuchungen über die Grundlagen der Mengenlehre (1908) by Ernst Zermelo. For example, in 1906, Gerhard Hessenberg used the name Menge der Teilmengen, i.e. set of subsets.

The German word Potenz comes form the Latin word potentia, which means power in English, and was used in different contexts, e.g. in politics and philosophy, but also in mathematics. So it probably was a combination of the fact what I mentioned above and that the power set is bigger, so more powerful.