## FANDOM

10,837 Pages

2 important things:

1) SGH catch-up ordinal is ψ_0(Ω_ω)

2) limit of BEAF/BEAN is at least θ(Ω^{Ω+1})

## Up to ω^ω

FGH SGH Hardy HAN BEAF BAN xE# Dollar Chained Arrows
2 ω^ω ω^2 n! {n,n} {n,n} E(n)n n$[1] n → n 3 ε_0 ω^3 n!1 {n,n,2} {n,n,2} E(n)n#n n$[2]

n → n → 2

k φ(k,0) ω^k n!(k-2) {n,n,k-1} {n,n,k-1} E(n)n##k n$[k] n → n → k-1 ω φ(ω,0) ω^ω n![1] {n,n,n} {n,n,n} E(n)n##n n$[[0]] n → n → n
ω+1 φ(1,0,0) ω^{ω+1} n![2] {n,n,1,2} {n,n,1,2} E(n)n##n#n n$[1[0]] n → n → n → 2 ω+2 φ(1,1,0) ω^{ω+2} n![3] {n,n,2,2} {n,n,2,2} E(n)n##n#n#n n$[2[0]] n → n → n → 3
ω+k φ(1,k-1,0) ω^{ω+k} n![k+1] {n,n,k,2} {n,n,k,2} E(n)n##n##k n$[k[0]] n → n → n → k ω2 φ(1,ω,0) ω^{ω2} n![1,2] {n,n,n,2} {n,n,n,2} E(n)n##n##n n$[[0][0]] n → n → n → n
ω2+1 φ(2,0,0) ω^{ω2+1} n![2,2] {n,n,1,3} {n,n,1,3} E(n)n##n##n#n n$[[0][0]1] n → n → n → n → 2 ω3 φ(2,ω,0) ω^{ω3} n![1,3] {n,n,n,3} {n,n,n,3} E(n)n##n##n##n n$[[0][0][0]] n →_2 5
ω k φ(k-1,0,0) ω^{ω k} n![1,k] {n,n,n,k} {n,n,n,k} E(n)n###k n$[[0][0]...[0][0]] k [0]'s n →_2 k+2 ω^2 φ(ω,0,0) ω^{ω^2} n![1,1,2] {n,n,n,n} {n,n,n,n} E(n)n###n n$[[1]] n →_2 n
ω^2+1 φ(1,0,0,0) ω^{ω^2+1} n![2,1,2] {n,n,1,1,2} {n,n,1,1,2} E(n)n###n#n n$[1[1]] n →_2 n →_2 2 ω^2+ω φ(1,0,ω,0) ω^{ω^2+ω} n![[1],1,2] {n,n,n,1,2} {n,n,n,1,2} E(n)n###n##n n$[[0][1]] n →_2 n →_2 n
ω^2+ω2 φ(1,1,0,0) ω^{ω^2+ω2} n![[1,2],1,2] {n,n,n,2,2} {n,n,n,2,2} E(n)n###n##n##n n$[[0][0][1]] n →_2 n →_2 n →_2 n ω^2+ω k φ(1,k-1,0,0) ω^{ω^2+ω k} n![[1,k],1,2] {n,n,n,k,2} {n,n,n,k,2} E(n)n###n###k n$[[0][0]...[0][0][1]] k [0]'s n →_3 k+2
ω^2*2 φ(1,ω,0,0) ω^{ω^22} n![1,2,2] {n,n,n,n,2} {n,n,n,n,2} E(n)n###n###n n$[[1][1]] n →_3 n ω^2*k φ(k-1,0,0,0) ω^{ω^2k} n![1,k,2] {n,n,n,n,k} {n,n,n,n,k} E(n)n####k n$[[1][1]...[1][1]] k [1]'s n →_{k+1} n
ω^3 φ(ω,0,0,0) ω^{ω^3} n![1,1,3] {n,n,n,n,n} {n,n,n,n,n} E(n)n####n n$[[2]] n →_X n ω^32 φ(1,ω,0,0,0) ω^{ω^3*2} n![1,2,3] {n,n,n,n,n,2} {n,n,n,n,n,2} E(n)n####n####n n$[[2][2]] n →_{X2} n
ω^3k φ(k-1,0,0,0,0) ω^{ω^3*k} n![1,k,3] {n,n,n,n,n,k} {n,n,n,n,n,k} E(n)n#####k n$[[2][2]...[2][2]] k [2]'s n →_{X^2} k ω^4 φ(ω,0,0,0,0) ω^{ω^4} n![1,1,4] {n,n,n,n,n,n} {n,n,n,n,n,n} E(n)n#####n n$[[3]] n →_{X^2} n
ω^k θ(Ω^{k-1}) ω^{ω^k} n![1,1,k] {n,k+2(1)2} {n,k+2[2]2} E(n)n#\text{^}#k n$[[k]] n →_{X^k} n ω^ω θ(Ω^ω) ω^{ω^ω} n![1,1,1,2] {n,n(1)2} {n,n[2]2} E(n)n#\text{^}#n n$[[[0]]] n →_{X^X} n

## Extension of Chained Arrows

Chained Arrows extended similiar to BEAF

All rules:

1. # → 1 → #\ = #

2. a = a

3. a →_1 b = a^b

4. # → a → b+1 = # → (# → (...(# → (#) → b)...) → b) → b

5. a →_{c+1#} b = a →_{c#} a →_{c#} a ... a →_{c#} a →_{c#} a

6. a →_{X#} b = a →_{b#} a

## From ω^ω to ε_0

FGH Hardy SGH HAN BEAF BAN E^ Dollar Chained Arrows
ω^ω ω^{ω^ω} θ(Ω^ω) [1,1,1,2] {n,n(1)2} {n,n[2]2} E(n)n#\text{^}#n [[[0]]] n →_{X^X} n
ω^ω+1 ω^{ω^ω+1} θ(Ω^ω+Ω) [2,1,1,2] {n,n,2(1)2} {n,n,2[2]2} E(n)n#\text{^}#n#n [1[[0]]] n →_{X^X} n →_{X^X} 2
ω^ω+ω+1 ω^{ω^ω+ω+1} θ(Ω^ω+Ω^2) [[2],1,1,2] {n,n,1,2(1)2} {n,n,1,2[2]2} E(n)n#\text{^}##n#n [1[0][[0]]] n →_{X^X} n →_{X^X} n →_{X^X} 2
ω^ω+ω2+1 ω^{ω^ω+ω2+1} θ(Ω^ω+Ω^2θ(Ω^ω+Ω^2)) [[2,2],1,1,2] {n,n,1,3(1)2} {n,n,1,3[2]2} E(n)n#\text{^}###n#n [1[0][0][[0]]]

n→_{X^X} n→_{X^X} n→_{X^X} n→_{X^X} 2

ω^ω+ω^2+1 ω^{ω^ω+ω^2+1} θ(Ω^ω+Ω^3) [[2,1,2],1,1,2] {n,n,n,n(1)2} {n,n,n,n[2]2} E(n)n#\text{^}####n [1[1][[0]]] n →_{X^X+1} n →_{X^X+1} 2
ω^ω2 ω^{ω^ω2} θ(Ω^ω2) [1,2,1,2] {n,n(1)3} {n,n[2]3} E(n)n#\text{^}#n#\text{^}#n [[[0]][[0]]] n →_{X^X2} n
ω^ω3 ω^{ω^ω3} θ(Ω^ω3) [1,3,1,2] {n,n(1)4} {n,n[2]4} E(n)n#\text{^}#n#\text{^}#n#\text{^}#n [[[0]][[0]][[0]]] n →_{X^X3} n
ω^{ω}k ω^{ω^{ω}k} θ(Ω^{ω}k) [1,k,1,2] {n,n(1)k+1} {n,n[2]k+1} E(n)n#\text{^}#*#k [[[0]][[0]]...[[0]][[0]]] k [[0]]'s

n →_{X^Xk} n

ω^{ω+1}+1 ω^{ω^{ω+1}} θ(Ω^{Ω}) [2,[1],1,2] {n,n,2(1)n} {n,n,2[2]n} E(n)n#\text{^}#*#n#n [1[1[0]]] n →_{X^{X+1}} n →_{X^{X+1}} 2
ω^{ω+1}2 ω^{ω^{ω+1}2}

θ(Ω^{Ω}2)

[1,[2],1,2] {n,n(1)n,2} {n,n[2]n,2} E(n)n#\text{^}#*#n#\text{^}#*#n [[1[0]][1[0]]] n →_{X^{X+1}2} n
ω^{ω+2} ω^{ω^{ω+2}} θ(Ω^{Ω+1}) [1,[1,2],1,2] {n,n(1)n,n} {n,n[2]n,n} E(n)n#\text{^}#*##n [[2[0]]] n →_{X^{X+2}} n
ω^{ω+k} ω^{ω^{ω+k}} θ(Ω^{Ω+k}) [1,[1,k],1,2] {n,n(1)(1)k} {n,n[2][2]k} E(n)n#\text{^}#*#\text{^}#k [[k[0]]] n →_{X^{X+k}} n
ω^{ω2} ω^{ω^{ω2}} θ(Ω^{Ω+ω}) [1,[1,1,2],1,2] {n,n(1)(1)n} {n,n[2][2]n} E(n)n#\text{^}#*#\text{^}#n [[[0][0]]] n →_{X^{X2}} n
ω^{ω3} ω^{ω^{ω3}} θ(Ω^{Ω2+ω}) [1,[1,2,2],1,2] {n,n(2)3} {n,n[3]3} E(n)n#\text{^}#*#\text{^}#n [[[0][0][0]]] n →_{X^{X3}} n
ω^{ω k} ω^{ω^{ω k}} θ(Ω^{Ω(k-1)+ω}) [1,[1,k-1,2],1,2] {n,n(2)k} {n,n[3]k} E(n)n#\text{^}##k

[[[0][0]...[0][0]]]

k [0] 's

n →_{X^{Xk}} n
ω^{ω^2} ω^{ω^{ω^2}} θ(Ω^{Ωω}) [1,[1,1,3],1,2] {n,n(2)n} {n,n[3]n} E(n)n#\text{^}##n [[[1]]] n →_{X^{X^2}} n
ω^{ω^22} ω^{ω^{ω^22}} θ(Ω^{Ω^2+ω^2}) [1,[1,2,3],2,2] {n,n(2)(2)n} {n,n[3][3]n} E(n)n#\text{^}##*#\text{^}##n [[[1][1]]] n →_{X^{X^2}} n
ω^{ω^2k} ω^{ω^{ω^2k}} θ(Ω^{Ω^2(k-1)}) [1,[1,k-1,3],1,2] {n,n(3)k} {n,n[4]k} E(n)n#\text{^}###k [[[1][1]...[1][1]]] k [1] 's n →_{X^{X^22}} n
ω^{ω^3} ω^{ω^{ω^3}} θ(Ω^{Ω^2ω}) [1,[1,1,4],1,2] {n,n(3)n}

{n,n[4]n}

E(n)n#\text{^}###n [[[2]]] n →_{X^{X^3}} n
ω^{ω^k} ω^{ω^{ω^k}} θ(Ω^{Ω^{(k-1)}ω}) [1,[1,1,k+1],1,2] {n,n(k)n} {n,n[k+1]n} E(n)n#\text{^}#\text{^}#k [[[k]]] n →_{X^{X^k}} n
ω^{ω^ω} ^4ω θ(Ω^{Ω^{ω}}) [1,1,2,2] {n,n(0,1)n} {n,n[1,2]n} E(n)n#\text{^}#\text{^}#n [[[[0]]]] n →_{X^{X^X}} n
^4ω ^5ω θ(Ω^{Ω^{Ω^ω}}) [1,1,2,3] {n,n((1)1)n} {n,n[1[2]2]n} E(n)n#\text{^}#\text{^}#\text{^}#n [[[[[0]]]]] n →_{X^{X^{X^X}}} n
^kω ^{k+1}ω θ(^{k}Ω) [1,1,2,k-1] X \uparrow\uparrow k \text{&} n {n,k[1/2]2} E(n)n#\text{^}\text{^}#k

[[...[[0]]...]]

k brackets

n →_{X → k → 2} n

ε_0

ε_0

θ(ε_{Ω+1}) [1,1,1,1,2] X \uparrow\uparrow n \text{&} n {n,n[1/2]2} E(n)n#\text{^}\text{^}#n [0]_2 n →_{X → X → 2} n

## From ε_0 to \Gamma_0

FGH Hardy HAN BEAF BAN xE^ and E{} Dollar

Chained Arrows

ε_0 ε_0 [1,1,1,1,2] X \uparrow\uparrow n \text{&} n {n,n[1/2]2} E(n)n#{##}#n [0]_2 n →_{X → X → 2} n
ε_0+1 ω^{ε_0+1} [2,1,1,1,2] {n,n,2(X \uparrow\uparrow X)2} {n,n,2[1/2]2} E(n)n#{##}#n#n [[0]_2] n →_{X → X → 2} n →_{X → X → 2} 2
ω^{ε_0+1} ω^{ω^{ε_0+1}} [1,[1],1,1,2] {n,n(X \uparrow\uparrow X)n} {n,n[1/2]n} E(n)n#{##}#*#n [[[0]_2]] n →_{X → X+1 → 2} n
ω^{ω^{ε_0+1}} ω^{ω^{ω^{ε_0+1}}} [1,1,[1],1,2] {n,n(X \uparrow\uparrow X*X)n} {n,n[2/2]2} E(n)n#{##}#\text{^}#n [[[[0]_2]]] n →_{X → X+2 → 2} n
ε_1 ε_1 [1,1,[1,1,2],1,2] {n,n(X \uparrow\uparrow X2)n} {n,n[1/3]2} E(n)n#{##}##n [1]_2 n →_{X → X2 → 2} n
ε_ω ε_ω [1,1,[1,1,3],1,2] X \uparrow\uparrow X^2 \text{&} n {n,n[1/1,2]2} E(n)n#{##}#\text{^}#n [[0]]_2 n →_{X → X^2 → 2} n
ε_{ω^ω} ε_{ω^ω} [1,1,[1,1,1,2],1,2] X \uparrow\uparrow X^X \text{&} n {n,n[1/1[2]2]2} E(n)n#{##}#\text{^}#\text{^}#n [[[0]]]_2 n →_{X → X^X → 2} n
ε_{ε_0} ε_{ε_0} [1,1,1,2,2] X \uparrow\uparrow X \uparrow\uparrow X\text{&} n {n,n[1/1[1/2]2]2} E(n)n#{##}#{##}#n [[0]_2]_2 n →_{X → 3 → 3} n
\zeta_0 \zeta_0 [1,1,1,1,1,2] X \uparrow\uparrow \uparrow X\text{&} n {n,n[1/1/2]2} E(n)n#{###}#n [0]_3 n →_{X → X → 3} n

\zeta_1

\zeta_1 [1,1,1,[1,1,2],1,2] X \uparrow\uparrow \uparrow X2\text{&} n {n,n[1/1/3]2} E(n)n#{###}##n [1]_3 n →_{X → X2 → 3} n

\zeta_{\zeta_0}

\zeta_{\zeta_0}

[1,1,1,1,2,2]

X \uparrow\uparrow \uparrow X \uparrow\uparrow \uparrow X\text{&} n {n,n[1/1[1/1/2]2]2} E(n)n#{###}#{###}#n [[0]_3]_3 n →_{X → 3 → 4} n
\eta_0 \eta_0 [1,1,1,1,1,1,2] {X,X,4}\text{&} n {n,n[1/1/1/2]2} E(n)n#{####}#n [0]_4 n →_{X → X → 4} n
φ(ω,0) φ(ω,0) [1]w/[1] {X,X,X}\text{&} n {n,n[1[1\sim2]2]2} E(n)n#{#\text{^}#}#n [0]_{[0]} n →_{X → X → X} n
φ(ω+1,0) φ(ω+1,0) [1]w/[2] {X,X,X+1}\text{&} n {n,n[1[2\sim2]2]2} E(n)n#{#\text{^}##}#n [0]_{[0]1} n →_{X → X → X+1} n
φ(ε_0,0) φ(ε_0,0) [1]w/[1,1,1,1,2] {X,X,{X,X,2}}\text{&} n {n,n[1[1/1\sim2]2]2} E(n)n#{#{##}#}#n [0]_{[0]_2} n →_{X → X → (X → X → 2) }n
φ(φ(ω,0),0) φ(φ(ω,0),0) [1]w/[1]w/[1] {X,X,{X,X,X}}\text{&} n

{n,n[1[1[1\sim2]2\sim2]2]2}

E(n)n#{#{#\text{^}#}#}#n [0]_{[0]_{[0]}} n →_{X → X → 2 → 2}n
φ(φ(φ(...,0),0),0)  k-1 φ's

φ(φ(φ(...,0),0),0)

k-1 φ's

[1]w/[1]w/[1]...[1]w/[1]w/[1]

k w/ 's

{X,k,1,2}\text{&} n

{n,n[1[1[1[1...2\sim2]2\sim2]2\sim2]2]2}

k \sim's

E(n)n#{#{#....#}#}#n

k nested layers

[0]_{[0]_{[0]_{...}}}

k+1 nested layers

n →_{X → X → k → 2}n

\Gamma_0

\Gamma_0 [1(1)2] {X,X,1,2}\text{&} n {n,n[1[1/2\sim2]2]2} E(n)n#{{#}}#n [0,1] n →_{X → X → X → 2}n

## From φ(1,0,0) to the LVO

FGH Hardy HAN BEAF BAN xE{} Dollar

φ(1,0,0)

φ(1,0,0)

[1(1)2] {X,X,1,2}\text{&} n {n,n[1[1/2\sim2]2]2} E(n)n#{{#}}#n [0,1]
φ(1,0,0)+1 φ(1,0,0)ω^ω [2(1)2] {n,n,2({X,X,1,2})2} {n,n,2[1[1/2\sim2]2]2} E(n)n#{{#}}#n#n [1,1]
φ(1,0,1) φ(1,0,1) [1(1)3] {X,X2,1,2}\text{&}n {n,n[1[1/2\sim2]3]2} E(n)n#{{##}}#n [0,2]
φ(1,0,k) φ(1,0,k) [1(1)k+2] {X,Xk+1,1,2}\text{&}n {n,n[1[1/2\sim2]k+2]2} E(n)n#{{#^k}}#n#n [0,k]
φ(1,1,0) φ(1,1,0) [1(1)[_21]] {X,X,2,2}\text{&}n {n,n[1[1/2\sim2]1/2]2} E(n)n#{{{#}}}#n#n [0,[0,1]_2]
φ(2,0,0) φ(2,0,0) [1(1)[_21,1,2]] {X,X,1,3}\text{&}n {n,n[1[1/2\sim2]1[1/2\sim2]2]2} limit of xE{} [0,0,1]
φ(1,0,0,0) φ(1,0,0,0) [1(1)[_21,[_21,1,3],1,2]] {X,X,1,1,2}\text{&}n {n,n[1[1/3\sim2]1]2} [0,0,[0,0,1]_2]
θ(Ω^{ω}) θ(Ω^{ω}) [1(1)[_21,[_21,1,[1]],1,2]] {X,X(1)2}\text{&}n {n,n[1[1/1,2\sim2]1]2} [0,0,[0,0,1]_{[0]}]
θ(Ω^{θ(Ω^{ω})}) θ(Ω^{θ(Ω^{ω})})

[1(1)[_21,[_21,1,[_21,1,[1]]],1,2]]

{X,X(1)2}\text{&}X\text{&}n {n,n[1[1/1[1[1/1,2\sim2]1]2\sim2]1]2} [0,0,[0,0,1]_{[0,0,1]_{[0]}}]
θ(Ω^{Ω}) θ(Ω^{Ω}) [1(1)[_21,1,2,2]] {n,n/2} {n,n[1[1/1/2\sim2]2]2} [0,0,0,1]

## Relationship between SGH and FGH

f_\alpha(n) \approx g_{θ(\beta)}(n)

f_{\alpha+1}(n) \approx g_{θ(\beta+Ω)}(n)

f_{\alpha+2}(n) \approx g_{θ(\beta+Ω2)}(n)

f_{\alpha+ω+1}(n) \approx g_{\beta+Ω^2)}(n)

f_{\alpha+ω^ω+1}(n) \approx g_{θ(\beta+Ω^ω)}(n)

f_{\alpha2}(n) \approx g_{θ(\beta2)}(n)

f_{\alphaω)}(n) \approx g_{θ(\betaω)}(n)

The thing is that when g_{θ(\beta)}(n) evaluates, n remains constant. So if you want to incresare a superscript, like in f_{ω^{ω+1})}(n) put ω+1 in the SGH subscript, won't help, it just evaluates to n+1. so you have to evaluate whole array so you have to put Ω in the SGH subscript.

Similiar, building a tower of ωs in FGH will build a tower of Ωs in SGH, building a tower of Ωs in FGH will build a tower of Ω_2s in SGH, etc.

So the paper is right and the catch-up ordinal is ψ_0(Ω_ω)

## Beyond LVO

FGH and Hardy HAN BEAF
θ(Ω^{Ω}) [1(1)[_21,1,2,2]] {n,n/2}

θ(Ω^{Ω})+1

θ(Ω^{Ω})ω^ω

[2(1)[_21,1,2,2]] {n,n,2/2}

θ(Ω^{Ω})2

θ(Ω^{Ω})^2

[1,2(1)[_21,1,2,2]] {n,n/3}

θ(Ω^{Ω})^2

θ(Ω^{Ω})^{θ(Ω^{Ω})}

[1,1,2(1)[_21,1,2,2]] {n,n/1/2}

θ(Ω^{Ω})^{θ(Ω^{Ω})}

^{3}θ(Ω^{Ω})

[1,1,1,2(1)[_21,1,2,2]] {X,X/2}\text{&&}n

ε_{θ(Ω^{Ω})+1}

[1,1,1,1,2(1)[_21,1,2,2]] {n,n//2}
\zeta_{θ(Ω^{Ω})+1} [1,1,1,1,1,2(1)[_21,1,2,2]] {L,L,2}_{n,n}
\eta_{θ(Ω^{Ω})+1} [1,1,1,1,1,1,2(1)[_21,1,2,2]] {L,L,3}_{n,n}
\Gamma_{θ(Ω^{Ω})+1} [1(1)[_22,1,2,2]] {L,L,L}_{n,n}
θ(Ω^{Ω}+1) [1(1)[_21,[1],2,2]] {L2,2}_{n,n}
θ(Ω^{Ω}+2) [1(1)[_21,[1,2],2,2]] {L3,2}_{n,n}
θ(Ω^{Ω}+ω) [1(1)[_21,[1,[1]],2,2]] {LX,2}_{n,n}
θ(Ω^{Ω}+ω^2) [1(1)[_21,[1,1,3],2,2]] {LX^2,2}_{n,n}
θ(Ω^{Ω}2) [1(1)[_21,1,3,2]] {LL,2}_{n,n}
θ(Ω^{Ω}ω) [1(1)[_21,1,[1],2]]

{X\text{&}L,2}_{n,n}

θ(Ω^{Ω}ω^2) [1(1)[_21,1,[1,2],2]] {X^2\text{&}L,2}_{n,n}
θ(Ω^{Ω}θ(Ω^{Ω})) [1(1)[_21,1,[1(1)[_21,1,2,2],2]] {{X,X/2}\text{&}L,2}_{n,n}
θ(Ω^{Ω}θ(Ω^{Ω}+ω)) [1(1)[_21,1,[1(1)[_21,1,[1,[1]],2],2]] {{L100,2}_{n,n}\text{&}L,2}_{n,n}
θ(Ω^{Ω+1}) [1(1)[_21,1,1,3]] limit?