1. Show, assuming AC, that any Cartesian product of finite sets is either finite or uncountable (can't be countably infinite).
2. Let X be the collection of all intervals [a, b] of length 0 ≤ b - a ≤ 2, with inclusion as partial ordering. Show that every chain in X has an upper bound.
3. In the application of the recursion principle 1.3.2 in the proof that WO implies HMP, how should g be defined? Show that the resulting argument does prove that WO implies HMP.