This is just imposing an chosen (and in this case, pretty natural) order on a set without total order. But as you note, you have to give something up. In this

This is just imposing an chosen (and in this case, pretty natural) order on a set without total order. But as you note, you have to give something up. In this case the order identifies each number on a circle in the complex plane with a single value, their magnitude. I've never really seen this notation used, but it could have some use in partially-ordered sets.