User blog:Alemagno12/yet another BMS analysis

Since BMS is becoming a popular topic now, why not have another try at an analysis?

(This'll probably be split into parts if it gets long enough)

As always, [WIP]

Up to ε0
The blank sequence corresponds to 0. Then, adding (0)s increases the ordinal by 1. (1) takes the limit of duplicating the (0) to its left. More specifically, (1) takes the limit of duplicating the (0)-type separator group to its left, which is just the rightmost (0) to its left and all elements to the right of that, excluding, of course, the (1). [WIP]
 * (0) = 1
 * (0)(0) = 2
 * (0)(0)(0) = 3
 * (0)(1) = (0)(0)(0)... = ω
 * (0)(1)(0) = ω+1
 * (0)(1)(0)(0) = ω+2
 * (0)(1)(0)(1) = (0)(1)(0)(0)(0)... = ω2
 * (0)(1)(0)(1)(0) = ω2+1
 * (0)(1)(0)(1)(0)(1) = ω3
 * (0)(1)(0)(1)(0)(1)(0)(1) = ω4
 * (0)(1)(1) = (0)(1)(0)(1)... = ω2
 * (0)(1)(1)(0) = ω2+1
 * (0)(1)(1)(0)(1) = ω2+ω
 * (0)(1)(1)(0)(1)(0)(1) = ω2+ω2
 * (0)(1)(1)(0)(1)(1) = (0)(1)(1)(0)(1)(0)(1)... = ω22
 * (0)(1)(1)(0)(1)(1)(0)(1) = ω22+ω
 * (0)(1)(1)(0)(1)(1)(0)(1)(1) = ω23
 * (0)(1)(1)(1) = (0)(1)(1)(0)(1)(1)... = ω3
 * (0)(1)(1)(1)(0)(1)(1) = ω3+ω2
 * (0)(1)(1)(1)(0)(1)(1)(1) = ω32
 * (0)(1)(1)(1)(1) = (0)(1)(1)(1)(0)(1)(1)(1)... = ω4
 * (0)(1)(1)(1)(1)(1) = ω5
 * (0)(1)(1)(1)(1)(1)(1) = ω6