[summary: A finite set is a set which is not infinite. That is, we could go through its members one by one, seeing a new member every minute, writing a mark on a piece of paper each time, and eventually we'd be done and stop writing.]
[summary(Technical): A finite set $~$X$~$ is a set which is not infinite: that is, there is some $~$n \in \mathbb{N}$~$ such that the Cardinality of $~$X$~$ is equal to $~$n$~$. Examples: $~$\{ 1,2 \}$~$ and $~$\{ \mathbb{N} \}$~$. Non-examples: $~$\mathbb{N}$~$, $~$\mathbb{R}$~$.]
A finite set is like a package for a group of things. The package can be monstrously big to fit the amount of things it needs to hold but ultimately there is a limit to the amount of items in the package. This means that if you were to go by each item in this package one by one and count them you would eventually reach the last item. (Incidentally this number that you counted would be the cardinality of the set)
[todo: Create a diagram expressing what a finite set is.]
[todo: examples, being the easiest way to understand what "finite set" means]
More formally a Finite set is a set that also has a bijection with one of the natural numbers.
[todo: the same as saying the cardinality is finite]
[todo: difference between "finite" and "hereditarily finite": in particular, {N} is finite though N is infinite]