User blog comment:MilkyWay90/A new ordinal notation/@comment-32213734-20180718180708

If we define α - 1 as (α - 1) + 1 = α, and if we define ordinal tetration similarly to ordinal addition, multiplication and exponentiation, that is


 * α ↑↑ 0 = 1
 * α ↑↑ (β + 1) = αα ↑↑ β
 * α ↑↑ β = sup(α ↑↑ β[n]) for limit β

then


 * N(0, 0) = ω (or maybe ω ↑↑ 0 = 1, according rule 2)
 * N(0, 1) = ω
 * N(0, 2) = ωω
 * N(0, 3) = ωω ω
 * N(0, ω) = ε0
 * N(0, ω + 1) = ωω ↑↑ ω = ωε0 = ε0
 * N(0, ω + 2) = ωω ↑↑ (ω + 1) = ωε0 = ε0
 * N(0, ω2) = ε0
 * N(0, infinite ordinal) = ε0
 * N(1, 0) = N(0, ω) = ε0
 * N(1, 1) = N(0, N(1, 0)) = N(0, ε0) = ε0
 * N(1, 2) = N(0, N(1, 1)) = N(0, ε0) = ε0
 * N(1, finite ordinal) = ε0
 * N(1, ω) = N(0, N(1, ω - 1)), but there is no ordinal ω - 1. So, N(1, infinite ordinal) is nonsence.
 * N(2, 0) = N(1, ω) = N(0, N(1, ω - 1)). So, N(ordinal > 1, m) is nonsence.
 * N(1, ω) = N(0, N(1, ω - 1)), but there is no ordinal ω - 1. So, N(1, infinite ordinal) is nonsence.
 * N(2, 0) = N(1, ω) = N(0, N(1, ω - 1)). So, N(ordinal > 1, m) is nonsence.

So, limit of this notation is ε0.