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Things I noticed:
This is a consequence of the fact that the power set of a set is always larger than the set itself (even for infinite sets!). We can tell via a one-to-one correspondence: two sets are the same size if a one-to-one correspondence exists between them.
Note that we sometimes show that one set is the same size as another by noting that one set (A) is a subset of the other (B), but that there are enough elements in A to cover B (this is the message of the Motel $\infty$). We frequently had rooms left over -- natural numbers left over -- even as we covered sets that contained the natural numbers (e.g. the positive rational numbers -- that is, the naturally numbered busses with naturally numbered seats: bus/seat).
We showed (by contradiction) that no such correspondence exists between a set and its power set. Hence \[ Card(N)< Card(P(N))< Card(P(P(N)))< \ldots \]
If you check out this code, you'll see where I compute the power set ("all-subsets") of a set, to cover all the bases. I think that the end result is pretty beautiful, too!
(coded up following our author's algorithm, in section 3.3, p. 214, as closely as possible).
