Infinity Scrapers

This is the summary of six fastest growing functions I built, using existing fast growing functions, nested ordinals and notation ↥ . Enjoy!

Part I: Nest Ordinals with Up Arrows

1.1- For ordinals ɑ, β, γ, δ and κ; positive integers a, b, c and d:

1. The basic form is a↑^ɑb and always expand from right to left.
2. β↑γ# = β^γ# (γ# denotes expressions of γ expansion in +, * and ^), e.g.
1. β↑(γ+1) = β^γ+1. 
2. β↑(γ²+γ) = β^γ2+γ = β^γ2*β^γ.
3. β↑^c+1b = β↑^cβ↑^c ... β↑^cβ (with b copies of β).
4. a↑^ɑ+1b = a↑^ɑa↑^ɑ... a↑^ɑa (with b copies of a's).
5. β↑^ɑ+1b = β↑^ɑβ↑^ɑ... β↑^ɑβ (with b copies of β's).
6. a↑^β↑ɑγ+1b = a↑ ^{...(((β↑ɑγ)↑ɑγ)↑ɑγ)...} b (with b copies of ↑^ɑγ 's).
7. a↑^{κ↑δβ↑ɑγ+1}b = a↑^{κ↑δ ...(((β↑ɑγ)↑ɑγ)↑ɑγ)...} b (with b copies of ↑^ɑγ 's).
8. All other rules shall follow the rules of Up Arrow and the rules of ordinal collapsing.
9. Define T(δ,d) = ...δ↑^δ↑δδδ... (with d floors; both δ and d are user-defined).

Examples:

1. 2↑^ω3 = 2↑³3 = ...
2. 2↑^ω+13 = 2↑^ω(2↑^ω2) = 2↑^ω(2↑²2) = 2↑^ω4 = 2↑⁴4 = ...
3. 2↑^ω↑ωω3 = 2 ↑^{ω↑ω 3} 3 = 2 ↑^{ω↑3 3} 3 = 2 ↑^{ω↑2ω↑2ω} 3 = ...
4. 2↑^ω↑ωω+13 = 2↑ ^{...(((ω↑ωω)↑ωω)↑ωω)...} 3 = ...
5. 2↑^{ω↑ω↑ω+1ωω}3 = 2↑^{ω↑ω↑ω+133}3 = 2↑^{ω↑ω↑ωω↑ωω3}3 ...
6. 2↑^ε₀3 = 2↑^ωωω3 = ...
7. 2↑^T(ε₀,ω)3 = 2↑^T(ε₀,3)3 = 2↑^{ε₀↑ε₀↑ε₀ ε₀ ε₀} 3 = ...

Part II: Nest Ordinals with CEN

2.1- For ordinals ɑ, β and δ, positive integer a:

1. Assign {CEN} as any sumbols from CEN (Cascading e-Notation of Sbiis Saibian), i.e. #, #^#, #^#*#, &(1), etc.
2. ɑ{CEN}β is a nested ordinal.
3. Expand {CEN} when there is a positive integer at its right; expand ordinals from right to left, e.g.
1. ɑ#a = ɑ^ɑ..ɑɑ (with a floors).
2. ɑ#β#a = ɑ#(ɑ#(...(ɑ#(ɑ#β)...)) (with a copies of ɑ's).
3. ɑ##a = ɑ#ɑ#...ɑ#ɑ (with a copies of ɑ's).
4. ɑ{CEN}β+1 = ... (((ɑ{CEN}β){CEN}β){CEN}β) ...
5. κ{CEN}ɑ{CEN}β+1 = κ{CEN}(...(((ɑ{CEN}β){CEN}β){CEN}β)...)
6. All other rules shall follow the rules of CEN and the rules of ordinal collapsing.
7. Define nested ordinal U(γ,δ) = γ&(1)δ = γ{{{...{{{#,#+1,1,2},#+1,1,2},#+1,1,2}...},#+1,1,2},#+1,1,2}δ; and U(γ,γ) = U(γ).

Examples:

1. 2↑^ω#ω3 = 2↑^ω#33 = 2↑^ωωω3 = ...
2. 2↑^{ω#^#ω+1}3 = 2↑^{(((ω#^#ω)#^#ω)#^#ω)}3 = ...

Part III: ↥ Notation for Functions

3.1- For positive integers k and n, the notation is right-associative, i.e. it's expanded from right to left.

1. Define positive integer function g(n).
2. g(ω)↥n = gⁿ(n).
3. g(ω)↥^k+1n = g(ω)↥^kg(ω)↥^k ... g(ω)↥^kg(n).

Examples:

1. Define t(n) = g(ω)↥ⁿg(n) and g(n) = n+1, then 
1. t(2) = g(ω)↥²g(2) 
2. = g(ω)↥²3 
3. = g(ω)↥g(ω)↥g(3) 
4. = g(ω)↥g(ω)↥4 
5. = g(ω)↥g⁴(4) 
6. = g(ω)↥g(g(g(g(4)))) 
7. = g(ω)↥8
8. = 16

Part IV: ↥ Notation & Ordinals

4.1- Combine Part I, II and III to give very powerful ↥ notation.

Examples:

1. Define t(n) = g(ω)↥^ω↑↑ωg(n) and g(n) = n+1, then
1. t(1) = g(ω)↥^ω↑↑ωg(1)
2. = g(ω)↥^ω↑↑ω2
3. = g(ω)↥^ω↑↑22
4. = g(ω)↥^ωω2
5. = g(ω)↥^ω+22
6. = g(ω)↥^ω+1g(2)
7. = g(ω)↥^ω+13
8. = g(ω)↥^ωg(ω)↥^ωg(3)
9. = g(ω)↥^ωg(ω)↥⁴4
10. = g(ω)↥^ωg(ω)↥³g(ω)↥³g(ω)↥³g(4) = ...
2. Define t(n) = g(ω)↥^ω##ωg(n) and g(n) = n+1, then
1. t(2) = g(ω)↥^ω##ωg(2)
2. = g(ω)↥^ω##33
3. = g(ω)↥^ω#ω#ω3
4. = g(ω)↥^ω#ω#33
5. = g(ω)↥^ω#ωωω3
6. = g(ω)↥^ω#ωω33 = ...

Part V: Box Function & Ordinals

5.1- Box function [a](n) is another way of expressing sequential function f_a(n).

5.2- Define [a](n) then follow these rules:

1. [ɑ+1](n) = [ɑ]([ɑ](...[ɑ]([ɑ](n))...)) (with n copies of [ɑ]'s). 
2. [ɑ↑↑...↑↑β+1](n) = [... (((ɑ↑↑...↑↑β)↑↑...↑↑β)↑↑...↑↑β) ...](n) (with n copies of ↑↑...↑↑β 's).
3. [κ↑↑...↑↑ɑ↑↑...↑↑β+1](n) = [κ↑↑...↑↑ ...(((ɑ↑↑...↑↑β)↑↑...↑↑β)↑↑...↑↑β)...](n) (with n copies of ↑↑...↑↑β 's).
4. [ɑ{CEN}β+1](n) = [ ...(((ɑ{CEN}β){CEN}β){CEN}β)... ](n) (with n copies of {CEN}β 's).
5. [κ{CEN}ɑ{CEN}β+1](n) = [κ{CEN} ...(((ɑ{CEN}β){CEN}β){CEN}β)...](n) (with n copies of {CEN}β 's).
6. All other rules shall follow ordinal expansion rules.

Examples:

1. Define [0](n) = n+1 and [k+1](n) = [k]ⁿ(n), then
1. [ω+1](2)
1. = [ω]([ω](2))
2. = [ω]([2](2))
3. = [ω](8)
4. = [8](8) = ...
2. [ω#^#ω+1](2)
1. = [((ω#^#ω)#^#ω)](2)
2. = [((ω#^#ω)#^#2)](2) = ...

Part VI: Ton Function Family

6.1- Now these are real Infinity Scrapers:

mini ton function

1. mton(n) = F₈(SCG(⍵))↥^AF₈(SCG(n)), where A = T(Σ, F₈(SCG(⍵))).

ton function

1. ton(n) = [A](⍵)↥^A[A](n), where A = T(B, [B](⍵)).
2. [a+1](n) = [a](⍵)↥^B[a](n), where B = T(C, [a](⍵)).
3. [0](n) = F₈(SCG(⍵))↥^CF₈(SCG(n)), where C = T(Σ, F₈(SCG(⍵))).

Ton function

1. Ton(n) = [A](⍵)↥^A[A](n), where A = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^B(⍵)[a](n), where
1. B(b+1) = T(B(b), [a](⍵)); and
2. B(0) = T(C, [a](⍵)).
3. [0](n) = F₈(SCG(⍵))↥^CF₈(SCG(n)), where C = T(Σ, F₈(SCG(⍵))).

TON function

1. TON(n) = [A(⍵)](⍵)↥^A(⍵)[A(⍵)](n), where 
1. A(a+1) = T(A(a), [B(⍵)](⍵)); and
2. A(0) = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^B(⍵)[a](n), where
1. B(b+1) = T(B(b), [a](⍵)); and
2. B(0) = T(C(⍵), [a](⍵)).
3. [0](n) = F₈(SCG(⍵))↥^C(⍵)F₈(SCG(n)), where
1. C(c+1) = T(C(c), F₈(SCG(⍵))); and
2. C(0) = T(Σ, F₈(SCG(⍵))).

Mega TON function

1. MTON(n) = [A(⍵)](⍵)↥^U(A(⍵))[A(⍵)](n), where
1. A(a+1) = U(A(a)); and
2. A(0) = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^U(B(⍵))[a](n), where
1. B(b+1) = U(B(b)); and
2. B(0) = T(C(⍵), [a](⍵)).
3. [0](n) = F₈(SCG(⍵))↥^U(C(⍵))F₈(SCG(n)), where
1. C(c+1) = U(C(c)); and
2. C(0) = T(Σ, F₈(SCG(⍵))).

Giga TON function

2.1- {ɑ}(n) shall follow expansion rules of Box Function in Part V above, where ɑ denotes any ordinal or nested ordinals.

2.2- From here, Giga TON can be defined as:

1. GTON(n) = [A(⍵)](⍵)↥^U(A(⍵))[A(⍵)](n), where
1. A(a+1) = U(A(a)); and
2. A(0) = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^U(B(⍵))[a](n), where
1. B(b+1) = U(B(b)); and
2. B(0) = T(C(⍵), [a](⍵)).
3. [0](n) = {U(C(ω))}(SCG(⍵))↥^U(C(⍵)){U(C(ω))}(SCG(n)), where
1. C(c+1) = U(C(c)); and
2. C(0) = T(Σ, {Σ}(SCG(⍵))).

Note:

1. As ... {2}(n) >> {1}(n) >> {0}(n) > MTON(n), GTON(n) is at totally different league from the other five members of Ton Family.

Tera TON function

3.1- To go a step further from GTON, ☷b☷(n) is defined as:

1. ☷0☷(n) ≈ L_{...U(U(U(Σ)))...}(n) (derived from LNG(n), with n of U's); and
2. ☷b+1☷(n) ≈ L_{...U(U(U(Σ)))...}(n) (derived from ☷b☷(n), with n of U's).

3.2- From here, Tera TON can be defined as:

1. TTON(n) = [A(⍵)](⍵)↥^U(A(⍵))[A(⍵)](n), where
1. A(a+1) = U(A(a)); and
2. A(0) = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^U(B(⍵))[a](n), where
1. B(b+1) = U(B(b)); and
2. B(0) = T(C(⍵), [a](⍵)).
3. [0](n) = ☷U(C(ω))☷(SCG(⍵))↥^U(C(⍵))☷U(C(ω))☷(SCG(n)), where
1. C(c+1) = U(C(c)); and
2. C(0) = T(Σ, ☷Σ☷(SCG(⍵))).

Peta TON function

3.1- To go a step further from TTON, ◩b⬔(n) is defined as (refer to Part VI here):

1. ◩0⬔(n) ≈ L_{UC(C(C(...C(𝛀n2,0)...,0),0))}(n) (derived from LNG(n), with n of 0's); and
2. ◩b+1⬔(n) ≈ L_{UC(C(C(...C(𝛀n2,0)...,0),0))}(n) (derived from ◩b⬔(n), with n of 0's).

3.2- From here, Peta TON can be defined as:

1. PTON(n) = [A(⍵)](⍵)↥^U(A(⍵))[A(⍵)](n), where
1. A(a+1) = U(A(a)); and
2. A(0) = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^U(B(⍵))[a](n), where
1. B(b+1) = U(B(b)); and
2. B(0) = T(C(⍵), [a](⍵)).
3. [0](n) = ◩U(C(ω))⬔(SCG(⍵))↥^U(C(⍵))◩U(C(ω))⬔(SCG(n)), where
1. C(c+1) = U(C(c)); and
2. C(0) = T(Σ, ◩Σ⬔(SCG(⍵))).

Exa TON function

Please refer to here.