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**Class of all finite sets**

In a higher algebra book that I'm working through, the natural numbers are constructed in the following manner:-

Consider the class [itex]S[/itex] of all finite sets. Now, [itex]S[/itex] 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, [itex]S=S_1⋃S_2⋃S_3⋃....[/itex]where [itex]S_1,S_2,S_3,[/itex] etc are disjoint equivalence classes, and to [itex]S_n[/itex], we give the label of the [itex]n[/itex]th natural number. This is how the natural numbers are constructed.

Now, as I understand it, the number of elements of [itex]S[/itex] for any [itex]n[/itex], has to be infinite. For instance, the number [itex]5[/itex] is the label given to [itex]S_5[/itex]. But [itex]5[/itex] can be represented in an infinite number of ways: five chairs, tables, coins, pencils, pens, etc. So, this means that [itex]S_5[/itex] is an infinite class, and so is any [itex]S_n[/itex].

The only way I see this possible is, the class [itex]S[/itex] 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?

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