Let’s enjoy some familiar concepts in sets and be happy with the exponential notation of function space.
Normally, a function from set X to set Y is represented as . Sometimes, people say that it is the exponential object in the category of Set, but what does it mean? Is it similar to the exponentiation of numbers, e.g. ?
We can view the disjoint union and intersection as addition and multiplication respectively. For example, we have
where denotes the two-pointed set. In face, the isomorphism is given by and where and . In general, it means that
Using exponential notation, it reads that . It satisfies familiar equalities we learned, e.g.
- where is the empty set and is the one-pointed set.
- for any non-empty set .
With this notation, we can drop the subset notation since is equivalent to where becomes the characteristic function on . That is,
if or otherwise.
Moreover, the belonging relation can be viewed as a function from one-pointed set to , e.g. . Note that is a proposition in set theory but a function is a proposition in the meta-language.
Looking at the above discussion more carefully, we may suspect whether we can talk about set theory purely in terms of objects, morphisms, hom-set and so on. Well, indeed! Basically it is called topos theory and maybe I can say something about topos theory after I fully understand it …