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Here are the rules:

  • 0 < a (a ≠ 0)
  • N(a,0) > N(b,0) (a > b)
  • N(a,b) > N(a,c) (b > c)
  • Ωx > N(a,b) (b < Ωx ∧ a < Ωx+1)
  • Ωx = N(Ωx+1,0)
  • 0 = {}
  • N(a,b) = {a,b} ∪ {x|x ∈ N(c,d) ∧ ((d < b ∧ c = a) ∨ (d < N(a,b) ∧ c < a)) ∧ c,d < N(a,b)}
  • Ωx = {x|x < Ωx}

I'm not even sure if this notation forms a well-ordered system of ordinals. If it does, second system (this notation up to N(N(N(Ω3,1),0),0)) has a limit of Ψ(Ψ(Ψλ0(0),1)).

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