User blog:PsiCubed2/Formal definition of Letter Notation up to R for integer arguments

We'll use ordinal notation for this:

Let α,β,≤⌈₀ be ordinals, with β being a limit ordinal.

Let n,k be positive integers.

Define a function Λ : (⌈₀ ∪ {⌈₀})×ℕ → ℕ as:


 * Λ(α+1,1) = 10


 * Λ(1,n) = En = 10n


 * Λ(α+1,n+1) = Λ(α,Λ(α+1,n))


 * Λ(β,n) = Λ(β[n],10)

Where β[n] is the n-th member of the fundamental sequence of β, given as follows:


 * For β=β₁+β₂ (with β₂≤β₁) we have β[n] = β₁+(β₂[n])


 * For β=ω we have β[n] = n


 * For β=ωα+1 we have β[n] = ωα×n


 * For β=ωβ₁ (with β₁<β a limit ordinal) we have β[n] = ωβ₁


 * For β=φ(α+1,0) we have β[1]=φ(α,0), β[n+1]=φ(α,β[n])


 * For β=φ(α₁+1,α₂+1) we have β[1]=φ(α₁+1,α₂)+1, β[n+1]=φ(α₁,β[n])


 * For β=φ(β₁,0) (with β₁<β a limit ordinal) we have β[n+1]=φ(β₁[n],0)


 * For β=φ(β₁,α+1) (with β₁<β a limit ordinal) we have β[n+1]=φ(β₁[n],φ(β₁,α)+1)


 * For β=φ(α,β₁) (with β₁<β a limit ordinal) we have β[n+1]=φ(α,β₁[n])


 * For β=⌈₀ we have ⌈₀[1]=1, ⌈₀[n+1]=φ(⌈₀[n],0)

And using these definitions, we can now define:


 * Q1 = R1 = 10


 * Qn = Λ(ε₀,n) (for n>1)


 * Rn = Λ(⌈₀,n) (for n>1)


 * nQm = Λ(ω↑↑m,n) for n+m>2

And retroactively, for n>1:


 * En = Λ(1,n)


 * Fn = Λ(2,n)


 * Gn = Λ(3,n)


 * Hn = Λ(4,n)


 * Jn = Λ(ω,n)


 * Kn = Λ(ω+1,n)


 * Ln = Λ(ω+2,n)


 * Mn = Λ(ω×2,n)


 * Nn = Λ(ω2,n)


 * Pn = Λ(ωω,n)

If I've made any mistake (which is pretty likely) please let me know.





















