Nest Ordinals with Functions

This will enhance the ordinals expansion greatly as they are currenyly expanded based on multiplication and exponentiation only.

Part I: Up Arrow & Ordinals for Sequential Functions

1. The up arrow notation can be added to ordinals in the form of f_α↑^kβ(n), where f is FGH fundamental sequence function, where k and n are positive integers; α and β are ordinals.
2. Similar to expansion of ordinals, α↑^kβ is right associative (expanding from right to left).
3. f_α↑^kβ+1(n) = f_{...(((α↑^kβ)↑^kβ)↑^kβ)...}(n) (with n copies of ↑^k 's).
4. All other rules shall follow the rules of Up Arrow and the rules of ordinal expansion.

Examples:

1. f_ω↑ω(2) = f_ω↑2(2) = f_ω²(2) = ...
2. f_ω↑↑ω(2) = f_ω↑↑2(2) = f_ω^ω(2) = f_ω²(2) = ...
3. f_{ω↑↑(ω+1)}(2) = f_{((ω↑↑ω)↑↑ω)}(2) = f_{((ω↑↑ω)↑↑2)}((2) = ...
4. f_{ε0↑↑↑ε0}(2)
1. = f_{ε0↑↑↑ω^ω}(2)
2. = f_{ε0↑↑↑ω²}(2)
3. = f_{ε0↑↑↑ω2}(2)
4. = f_{ε0↑↑↑ω+ω}(2)
5. = f_{ε0↑↑↑ω+2}(2)
6. = f_{((ε0↑↑↑ω+1)↑↑↑ω+1)}(2)
7. = f_{(((ε0↑↑↑ω+1)↑↑↑ω)↑↑↑ω)}(2)
8. = ...
5. f_{φ(ω,0)↑↑ω}(2)
1. = f_{φ(ω,0)↑↑2}(2)
2. = f_{φ(ω,0)^φ(ω,0)}(2)
3. = f_{φ(ω,0)^φ(2,0)}(2)
4. = f_{φ(ω,0)^{φ(1,φ(1,0))}}(2)
5. = ...
6. f_{φ(ω,0)↑↑↑ω}(3)
1. = f_{φ(ω,0)↑↑↑3}(3)
2. = f_{φ(ω,0)↑↑φ(ω,0)↑↑φ(ω,0)}(3)
3. = f_{φ(ω,0)↑↑φ(ω,0)↑↑φ(3,0)}(3)
4. = f_{φ(ω,0)↑↑φ(ω,0)↑↑φ(2,φ(2,φ(2,0)))}(3)
5. = ...

Part II: Multi-nested Ordinals for Sequential Functions

1. In general form, for ordinals or nested ordinals α, γ and λ, positive integer a:
1. f_α↑^λγ(n) is right-associative.
2. f_α↑^λ+1a(n) = f_{α↑^λα↑^λ ... α↑^λα}(n) (with a copies of α's).
3. All other rules shall follow the rules of Part I above.

Part III: Non-Sequential Functions

1. The definition is similar to ordinals as subscripts, so I'll just go straight to examples.
2. For positive integer n and non-negative integer k, assign g(n) = n+1, then
1. g^ω(2) = g²(2) = g(g(2)) = 4.
2. g^ω+1(2) = g^ω(g^ω(2)) = g^ω(4) = g⁴(4) = 8. 
3. g^ω+k(n) = f_k+1(n) of FGH.
4. g^ϵ₀(2) = g^ωω(2) = g^ω2(2) = g^ω+2(2) = g^ω+1(g^ω+1(2)) = g^ω+1(8) = ... 
3. Obviously all ordinals are applicable and g(n) can be any non-sequential function, including G(n), n(k), TREE(n), SCG(n), Tar(n), D(k), Σ(n), etc.

Part IV: Nested Ordinals for Non-Sequential Functions

1. Nested Ordinal are basically the same as Part I and II above.

Examples

1- For positive integer n and non-negative integer k, assign g(n) = n+1, then

1. g^ω↑ωω+1(2) = g^{((ω↑ωω)↑ωω)}(2) = g^{((ω↑ωω)↑22)}(2) = ...

Part V: Chain of Nested Ordinals for All Functions

1. From Part I to IV above, a chain of nested ordinals can be defined as it is right associative.
2. For any ordinal or nested ordinal O, this is expressed as g_O^O@a(n) = g_O^O_O ... ^O_O^O(n) (with a pairs of _O^O 's, where a is positive integer).

Examples:

1- Using sequential function of FGH:

1. f_ω↑^ωω^ω(2) = f_ω↑^ωω²(2) = f_ω↑^ωω(f_ω↑^ωω(2)) = f_ω↑^ωω(f_ω↑²2(2)) = f_ω↑^ωω(f_ω^ω(2)) = ...

Part VI: Nest CEN with OCFs

1. General form is U_OCF(Expressions), where OCF (Ordinal Collapsing Function) can be Veblen's 𝝋, Buchholz's 𝜓, Rathjen's 𝜓, Taranovsky's C, etc. Here I only used Taranovsky's C as example, as it is the largest ordinal for computable functions.
2. The example below used Taranovsky's C, which is defined in here.
3. For Tar. C, C(1,0) = ⍵; for nested U system, U_C(1,0) = U(Σ).
4. For Tar. C, C(𝛀₁,0) = ε₀ = ⍵^⍵⍵...; for nested U system, U_C(𝛀₁,0) = (U(Σ))^{(U(Σ))(U(Σ))...}.
5. The largest ordinal for Tar. C is C(C(...C(𝛀_n2,0)...,0),0); for nested U system, it is U_C(C(C(...C(𝛀_n2,0)...,0),0)).