The definition does feel artificial on the edge case exclusion, but obviously for uniqueness it's necessary...

Kinda interesting why we don't have higher level concepts, as you say. Guessing you had this thought while toiling over the twin prime question.

Currently vibing on Surreal Numbers. A proper class, like the ordinals, but a generalization of all linear orders (instead of well orderings). Worth a gander if you're ever bored