Introduction

Welcome

Welcome to ordinal notations made by Aarex, but first, I will define the ordinal collapsing notation up to psi(α), where α is limit for Ω_a.

Ordinal Collapsing Notation

• NOTE: a_b is b subscript of a and a^b is b superscript of a.
• NOTE 2: # can be empty, any ordinal, or any cardinal.

FGH rules

• f_0(n) = n+1
• f_(α+1)(n) = f_(α)^n(n)
• f_α(n) = f_α[n](n) (If α isn't added by finite numbers.)

ω rule

For now on, I will use fundamental sequences.

• ω[n] = n

ε rules

ε is the form of ε(n)

• ε(0)[0] = 1
• ε(n)[0] = ε(n-1)+1
• ε(n)[m] = ω^ε(n)[m-1]

φ rules

φ is the form of φ(n,m)

• If n = 0 then:
  • φ(0,m) = ω^n
• If n > 0 then:
  • φ(n,0)[0] = 1
  • φ(n,m)[0] = φ(n,m-1)+1
  • φ(n,m)[x] = φ(n-1,φ(n,m)[x-1])

ψ and Ω rules

ψ is the form of ψ(α)

• ψ(0)[0] = 1
• ψ(α+1)[0] = ψ(α+1)+1
• ψ(α+1)[n] = ω^ψ(α+1)[n-1]
• ψ(α+Ω)[0] = 1
• ψ(α+Ω)[n] = ψ(α+ψ(α+Ω)[n-1])
• ψ(α)[n] = ψ(α[n])

Ω_α rules

ψ_n is the forms of ψ_β(α) and ψ_β(α+Ω_γ)

• If β = 0 then:
  • ψ_0(α) = ψ(α)
• If β > 0 then:
  • ψ_β(0)[0] = Ω_β+1
  • ψ_β(α+1)[0] = ψ(α+1)+1
  • ψ_β(α+1)[n] = ω^ψ(α+1)[n-1]
  • ψ_β(α+Ω_γ)[0] = 1
  • ψ_β(α+Ω_(γ+1))[n] = ψ_β(α+ψ_γ(α+Ω_(γ+1))[n-1])
  • ψ_β(α+Ω_γ)[n] = ψ_β(α+Ω_(γ[n]))
  • ψ_β(α)[n] = ψ_β(α[n])

Any mistakes? I hope there will be no mistakes.