So you define max([-∞, a]) =a, and define min(a, b) as max([-∞, a] ∩ [-∞, b])? Isn't this just shifting the problem elsewhere? For instance now you need to prov

So you define max([-∞, a]) =a, and define min(a, b) as max([-∞, a] ∩ [-∞, b])? Isn't this just shifting the problem elsewhere? For instance now you need to prove that the intersection of sets is of the form [-∞, a] or [-∞, b], which is fairly intuitive, but not more intuitive than the usual definition of min.