兩組函數之間的指數表示法歷史


2

眾所周知,如果 $ A $ $ B $ 是兩組,則從 $ A $ $ B $ 的所有函數的集合可以用 $ B ^ A $ :可以在許多地方找到對此特定表示法的解釋:

https://math.stackexchange.com/questions/901735/meaning-of-a-set-in-the-exponenthttps://math.stackexchange.com/questions/63960/what-does-it-mean-when-a-set-is-the-exponenthttps://math.stackexchange.com/questions/709184/why-is-the-exponential-of-sets-the-function-set

我要問的是:何時首次引入該符號?在什麼情況下使用?(因此,該問題並非關於其含義或背後的原理。)

我能找到的較舊的事件發生在1954年的Bourbaki的Théorie des ensembles,E.R.20,但這是第一次嗎?

3

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.