User blog:Alemagno12/How to intuitively understand BMS

NOTE: This blog post focuses on the behaviour of BMS rather than its ruleset. Also, this blog post uses BM2.

Primitive Sequence System
Think of it like this: This is basically an encoding of Cantor's normal form, and so the limit is ε0.
 * The empty string is 0.
 * (0) is 1.
 * Concatenation of strings of the form (0)X (X can be empty) is addition.
 * (0)X represents ωys for some y.
 * However, to represent y, we cannot use (0)Xs, so we use (1)Xs instead. (if y = 1, X = (1); if y = 2, X = (1)(1); etc.)
 * Then, (n+1)Xs are to (n)Xs as (1)Xs are to (0)Xs.

Pair Sequence System
Definition: An n-row BMS expression X is standard if there exists an x such that by evaluating (0,0,0,...,0)(1,1,1,...,1)[x] in (n+1)-row BMS (where f(n) = n), we can eventually get to X[x]. Property: In a standard n-row BMS expression, appending an entry at the start of each element of the sequence enumerating the elements always produces a standard expression in (n+1)-row BMS. [WIP]