In a higher algebra book that I'm working through, the natural numbers are constructed in the following manner:-
Consider the class $S$ of all finite sets. Now, $S$ is partitioned into equivalence classes based on the equivalence relation that two finite sets are equivalent if there exists a one-to-one correspondence between them, i.e. if they are equipotent. And each of these equivalence classes are given a label, corresponding to the number of one-to-one correspondences.
So, $S= S_1 \bigcup S_2 \bigcup S_3 \bigcup....$where $S_1, S_2, S_3,$ etc are disjoint equivalence classes, and to $S_n$, we give the label of the $n$th natural number. This is how the natural numbers are constructed.
Now, as I understand it, the number of elements of $S_n$ for any $n$, has to be infinite. For instance, the number $5$ is the label given to $S_5$. But $5$ can be represented in an infinite number of ways: five chairs, tables, coins, pencils, pens, etc. So, this means that $S_5$ is an infinite class, and so is any $S_n$.
The only way I see this possible is, the class $S$ that we started out with, has to be an infinite class. Is this true? Basically, what I'm asking is: Is the class of all finite sets infinite? How do you prove this?