User blog comment:MilkyWay90/Help with understanding Veblen array notation/@comment-30754445-20180811202716/@comment-31966679-20180818000734

Also, I was trying to calculate φ(1, 0, 0) when this happened:

By rule 4, φ(1, 0, 0) = sup{X_n} where X is:

X_1 = 0

X_(n + 1) = φ(0, X_n, 0, 0)

So trying to find the limit, I tried seeing a pattern by using X_1, then X_2, so then I could find a pattern and use X_omega:

X_2 = φ(0, 0, 0, 0) = φ (by rule 1) = undefined

And even if I use another way of calculating it:

X_2 = φ(0, 0, 0, 0) = φ(0) (by rule 1) = omega ^ 0 (rule 2) = 1

May seem like it worked, but:

X_3 = φ(0, X_2, 0, 0) = φ(0, 1, 0, 0) = φ(1, 0, 0) which is the number we started with, therefore it's an infinite loop!

What have I got wrong here?