# 兩組函數之間的指數表示法歷史

It comes from Bernstein's Habilitation dissertation Untersuchungen aus der Mengenlehre (1901, published 1905), where he also introduced the now common symbolism for cardinal arithmetic. The exponential notation is introduced in §2 as follows (my translation):

"If $$M$$ and $$N$$ are two sets, we call that set which - in the sense of a known expression - contains all combinations of elements from $$M$$ to the classes of $$N$$, the power $$M^N$$ ($$M$$ raised to $$N$$). Regarding the application to addition, multiplication and powerclasses of commutative and associative laws, they are the same as for finite numbers".

Bernstein does not use $$2^N$$ for the powerset, but he does write $$2^{\aleph_\alpha}$$, meaning set cardinality, in §9, when discussing the continuum hypothesis.