Everything's a Goat

I ran into an amusing site today, Goatism, in which the author maintains that the main principle of the life, the universe, and everything is that everything is a goat. That's right, a goat. You're a goat, I'm a goat. Ba-a-a-a. Bleat, bleat. The whole world's a goat. He has devoted some pages to the consequences of this. But his most interesting parts are when he proves that everything is a goat. His first proof has some obvious non-sequitors in it, so I will present a revision of his proof, and show that indeed, everything is a goat:

Theorem: Everything's a goat.

Lemma: If a set X contains a goat, then all elements of X are goats.

Proof: by induction on X = n. If n = 1, then if X contains a goat, then the only x in X is a goat, and since x is the only element of X, all elements of X are goats.

Suppose that for all Y with Y <= n, that if Y contains a goat, then all y in Y are goats. Let Zbe an n+1-element set with at least one goat g. Pick an h in Z<> g, and pull it out; i.e., consider Z- {h}. This set has n elements and it has g in it, so all elements of Z- {h} are goats.Remove another element k from the set, not g, and replace it with h. That is, consider Z - {k} union {h}. This set also has n elements, and contains g, so all elements in it are goats. This means that h is a goat also, so all elements of Z are goats.

There exists a goat g. I can take you to the zoo and prove that for you, or click here. Let a be anything, and consider {a,g}. This set has at least one goat in it, so it is all goats. Hence a is a goat. So anything is a goat. This completes the proof.

Note that I can also use this proof to prove everything is a leprechaun, since leprechauns do exist. I can use it to prove everyone's an orange, everyone's a genius, and everyone's anything. So where is the flaw in the proof? I say to remove another element k from the set, and that it can't be g. Of course not, since then I could not guarantee that there is a goat in the resulting set. But it also isn't h, since h has been pulled out. So the set out of which I pull my k has two elements less than the overall set, so it has number n - 1. However, when n+ 1 = 2, then n - 1 = 0. So the set out of which I have to pull my k has nothing in it, so I can't pull anything out of it. That destroys the entire proof, since we can't even proceed to n = 2 in the induction.

There are lots of neat little tricks on the site, such as a proof that if there is a goat, then there are not two goats, so the Entire Universe is One Big Goat. It's good for a laugh, but watch the math. There are traps ahead that will get your goat.