Lemma: All horses are the same color.
Proof (by induction):
Case n = 1: In a set with only one horse, it is obvious that all
horses in that set are the same color.
Case n = k: Suppose you have a set of k+1 horses. Pull one of these
horses out of the set, so that you have k horses. Suppose that all
of these horses are the same color. Now put back the horse that you
took out, and pull out a different one. Suppose that all of the k
horses now in the set are the same color. Then the set of k+1 horses
are all the same color. We have k true => k+1 true; therefore all
horses are the same color.
Theorem: All horses have an infinite number of legs.
Proof (by intimidation):
Everyone would agree that all horses have an even number of legs. It
is also well-known that horses have forelegs in front and two legs in
back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a
horse to have! Now the only number that is both even and odd is
infinity; therefore all horses have an infinite number of legs.
However, suppose that there is a horse somewhere that does not have an
infinite number of legs. Well, that would be a horse of a different
color; and by the Lemma, it doesn't exist.
