Scott Brickner wrote:
It's actually not arbitrary. [...] A ≤ B iff A ⊆ B A ⊆ B iff (x ∊ A) ⇒ (x ∊ B)
Alternatively and dually but equally naturally, A ≥ B iff A ⊆ B iff (x ∊ A) ⇒ (x ∊ B) and then we would have False > True. Many of you are platonists rather than formalists; you have a strong conviction in your intuition, and you call your intuition natural. You think ∅≤U is more natural than ∅≥U because ∅ has fewer elements than U. (Why else would you consider it unnatural to associate ≥ with ⊆?) But that is only one of many natural intuitions. There are two kinds of natural intuitions: disjunctive ones and conjunctive ones. The elementwise intuition above is a disjunctive one. It says, we should declare {0}≤{0,1} because {0} corresponds to the predicate (x=0), {0,1} corresponds to the predicate (x=0 or x=1), you see the latter has more disjuncts, so it should be a larger predicate. However, {0} and {0,1} are toy, artificial sets, tractible to enumerate individuals. As designers of programs and systems, we deal with real, natural sets, intractible to enumerate individuals. For example, when you design a data type to be a Num instance, you write down two QuickCheck properties: x + y = y + x x * y = y * x And lo, you have specified a conjunction of two predicates! The more properties (conjuncts) you add, the closer you get to ∅ and further from U, when you look at the set of legal behaviours. Therefore a conjunctive intuition deduces ∅≥U to be more natural.