Not Taranovsky's C

Here are the rules:

• 0 < a (a ≠ 0)
• N(a,0) > N(b,0) (a > b)
• N(a,b) > N(a,c) (b > c)
• Ω_x > N(a,b) (b < Ω_x ∧ a < Ω_x+1)
• Ω_x = N(Ω_x+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)}
• Ω_x = {x|x < Ω_x}

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(Ψ_λ0(0),1)).