This is an interesting article. But I think he's being too mysterious about this notion of "cheating". Consider a far smaller set: {1, 2, 3}. Bob wants to prov

This is an interesting article. But I think he's being too mysterious about this notion of "cheating". Consider a far smaller set: {1, 2, 3}. Bob wants to prove that there are at most 3 elements in this set. (Yes, I know this is trivial.) His strategy: name 3 elements in the set (1, 2, 3), and no matter what Alice picks from the set, he will name her choice. Obviously, this strategy works. Now suppose that Bob wants to prove that there is only 1 element in the set {1, 2, 3}. His strategy is similar: name 1 element in the set (2, say), and -- he hopes -- no matter what Alice picks from the set, he will name her choice. Of course, this doesn't work. However, in order to be sure that Bob does not name her choice: Alice must know his strategy. Does that mean she is "cheating"? Hardly. The above "cheating" issue is the same as that in the article. So we're not talking about some mysterious property of infinite sets.