User blog comment:Ubersketch/An outline for an ordinal notation/@comment-35470197-20190712053014

Your style to write down rule sets is ambiguous, because you always describe properties but not computation process or recursion. I explain how ambiguous it is.

> 0=#+1 where # is any string.

This equality means 0=+1 (# is the empty string), 0=0+1 (# is 0), 0=1+1 (# is 1), 0=0=+1 (# is 0=), and so on. It does not describe a computation process, a recursion, or a restriction of expressions.

> n+1 finds the leftmost n and replicates it infinitely (w times) and then gets killed.

Consider the expression 1,2,1 and n=1. This means 2 finds the leftmost 1 and replicates the 1 infinitely and then gets killed.

1,2,1 -> 1,1,…,2,1 (replicated at the right position of the 1) -> 1,1,…,1?

1,2,1 -> 1,2,1 1,1,… (replicated at an irrelevant side) -> 1,1 1,1,…?

1,2,1 -> …,1,1,2,1 (replicated at the left position of the 1) -> …,1,1,1?