BEAF (My Version) Beyond Tetrational Arrays

BEAF(My Version)

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## Explanation

### Up to ζ0

{a,b (0\1) 2} = {a,a ((((... ((((0,1)1)1)1)1)...1)1)1)1) 2} with b pairs of 1's. FGH level ε0.

{a,b,2 (0\1) 2} = {a,{a,{a...{a,{a,{a,a (0\1) 2} (0\1) 2} (0\1) 2}...(0\1) 2} (0\1) 2} (0\1) 2} (b times) FGH level ε0 + 1.

{a,b,1,2 (0\1) 2} = {a,{a,{a...{a,{a,{a,a,a (0\1) 2} (0\1) 2} (0\1) 2}...(0\1) 2} (0\1) 2} (0\1) 2} (b times) FGH level ε0 + ω.

{a,b (1) 2 (0\1) 2} = {a,a,a...a,a,a (0\1) 2} (b a's). FGH level ε0 + ω^ω.

{a,b (0,1) 2 (0\1) 2} = {a,a (b) 2 (0\1) 2}. FGH level ε0 + ω^ω^ω.

{a,b (0\1) 3} = {a,a ((((...((((0,1)1)1)1)1)...1)1)1)1) 2 (0\1) 2} (b pairs of 1's). FGH level (ε0)2.

{a,b (0\1) 4} = {a,a ((((...((((0,1)1)1)1)1)...1)1)1)1) 2 (0\1) 3} (b pairs of 1's). FGH level (ε0)3.

{a,b (0\1) 1,2} = {a,a ((((...((((0,1)1)1)1)1)...1)1)1)1) 2 (0\1) a} (b pairs of 1's). FGH level (ε0)ω.

{a,b (0\1)(1) 2} = {a,a (0\1) a,a,a...a,a,a} (b a's). FGH level (ε0)ω^ω.

{a,b (0\1)(0,1) 2) = {a,a (0\1)(b) 2}. FGH level (ε0)ω^ω^ω.

{a,b (0\1)(0\1)2} = {a,a (0\1)((((...((((0,1)1)1)1)1)...1)1)1)1) 2} (b pairs of 1's). FGH level ε0^2.

{a,b (0\1)(0\1)(0\1) 2} = {a,a (0\1)(0\1)((((...((((0,1)1)1)1)1)...1)1)1)1) 2} (b pairs of 1's). FGH level ε0^3.

{a,b (1\1) 2} = {a,a (0\1)(0\1)(0\1)...(0\1)(0\1)(0\1) 2} (b 0\1's). FGH level ε0^ω.

{a,b (2\1) 2} = {a,a (1\1)(1\1)(1\1)...(1\1)(1\1)(1\1) 2} (b 1\1's). FGH level ε0^ω^2.

{a,b (0,1\1) 2} = {a,a (b\1) 2}. FGH level ε0^ω^ω.

{a,b ((0\1)\1) 2} = {a,a (((((...((((0,1)1)1)1)1)...1)1)1)1)\1)2} (b pairs of 1's). FGH level ε0^ε0.

{a,b (((0\1)\1)\1) 2} = {a,a ((((((...((((0,1)1)1)1)1)...1)1)1)1)\1)\1) 2} (b pairs of 1's). FGH level ε0^ε0^ε0.

{a,b (0\2) 2} = {a,a ((((...((((0\1)\1)\1)\1)...\1)\1)\1)\1) 2) {b pairs of 0\1's). FGH level ε1.

{a,b ((0\2)\2) 2} = {a,a (((((...((((0,1)1)1)1)1)\2) 2} (b pairs of 1's). FGH level ε1^ε1.

{a,b (0\3) 2} = {a,a ((((...((((0\2)\2)\2)\2)...\2)\2)\2)\2) 2} (b pairs of 0\2's). FGH level ε2.

{a,b (0\0,1) 2} = {a,a (0\b) 2}. FGH level εω.

{a,b (0\1,1) 2} = {a,a (0\(0,1)(0,1)(0,1)...(0,1)(0,1)(0,1)) 2} (b (0,1)'s). FGH level ε(ω + 1).

{a,b (0\0,2) 2} = {a,a (0\b,1) 2}. FGH level ε(ω2).

{a,b (0\0,0,1) 2} = {a,a (0\0,b) 2}. FGH level ε(ω^2).

{a,b (0\(1)1) 2} = {a,a (0\0,0,0...0,0,0,1) 2} (b 0's). FGH level ε(ω^ω).

{a,b (0\(0,1)1) 2} = {a,a (0\(b)1) 2}. FGH level ε(ω^ω^ω).

{a,b (0\(0\1)) 2} = {a,a (0\((((...((((0,1)1)1)1)1)...1)1)1)1) 2} (b pairs of 1's). FGH level ε(ε0).

{a,b (0\(0\2)) 2} = {a,a (0\((((....((((0\1)\1)\1)\1)...\1)\1)\1)\1) 2} (b pairs of 0\1's). FGH level ε(ε1).

{a,b (0\(0\0,1)) 2} = {a,a (0\(0\b)) 2}. FGH level ε(εω).

(a,b (0\(0\(0\1))) 2} = {a,a (0\(0\((((...((((0,1)1)1)1)1)...1)1)1)1)) 2} (b pairs of 1's). FGH level ε(ε(ε0)).

{a,b (0\(0\(0\(0\1)))) 2} = {a,a (0\(0\(0\((((...((((0,1)1)1)1)1)...1)1)1)1))) 2} (b pairs of 1's). FGH level ε(ε(ε(ε0))).

### ζ0 - η0

{a,b (0\0\1) 2} = {a,a (0\(0\(0\...(0\(0\(0\(0\1))))...)))) 2} (b times). FGH level ζ0.

{a,b,2 (0\0\1) 2} = {a,{a,{a,{a,...{a,{a,{a,{a,a (0\0\1) 2} (0\0\1) 2} (0\0\1) 2} (0\0\1) 2}...(0\0\1) 2} (0\0\1) 2} (0\0\1) 2} (0\0\1) 2} (b times). FGH level ζ0 + 1.

{a,b (0\0\1) 3} = {a,a (0\(0\(0\...(0\(0\(0\(0\1))))...)))) 2 (0\0\1) 2} (b times). FGH level (ζ0)2.

{a,b (0\0\1)(0\0\1) 2} = {a,a (0\0\1)(0\(0\(0\...(0\(0\(0\(0\1))))...)))) 2} (b times). FGH level ζ0^2.

{a,b (1\0\1) 2} = {a,a (0\0\1)(0\0\1)(0\0\1)...(0\0\1)(0\0\1)(0\0\1) 2} (b times). FGH level ζ0^ω.

{a,b ((0\0\1)\0\1) 2} = {a,a ((0\(0\(0\...(0\(0\(0\(0\1))))...))))\0\1) 2} (b times). FGH level ζ0^ζ0.

{a,b (0\1\1) 2} = (a,a (((...(((0\0\1)\0\1)\0\1)...\0\1)\0\1)\0\1) 2} (b times). FGH level ε(ζ0 + 1).

{a,b (0\0,1\1) 2} = {a,a (0\b\1) 2}. FGH level ε(ζ0 + ω).

{a,b (0\(0\1)\1) 2} = {a,a (0\((((...((((0,1)1)1)1)...1)1)1)1)\1) 2} (b pairs of 1's). FGH level ε(ζ0 + ε0).

{a,b (0\(0\0\1)\1) 2} = {a,a (0\(0\(0\(0\...(0\(0\(0\(0\1))))...))))\1) 2} (b times). FGH level ε((ζ0)2).

{a,b (0\0\2) 2} = {a,a (0\(0\(0\...(0\(0\(0\0\1)\1)\1)...\1)\1)\1) 2} (b times). FGH level ζ1.

{a,b (0\0\0,1) 2} = {a,a (0\0\b) 2}. FGH level ζω.

{a,b (0\0\(0\1)) 2} = {a,a (0\0\((((...((((0,1)1)1)1)...1)1)1)1) 2} (b pairs of 1's). FGH level ζ(ε0).

{a,b (0\0\(0\0\1)) 2} = {a,a {0\0\(0\(0\(0\...(0\(0\(0\(0\1))))...)))) 2} (b times). FGH level ζ(ζ0).

{a,b (0\0\(0\0\(0\0\1))) 2} = {a,a (0\0\(0\0\(0\(0\(0\...(0\(0\(0\(0\1))))...)))) 2} (b times). FGH level ζ(ζ(ζ0)).

### η0 - φ(ω,0)

{a,b (0\0\0\1) 2} = {a,a (0\0\(0\0\(0\0\...0\0\(0\0\(0\0\1))))...)))) 2} (b times). FGH level η0 = φ(3,0).

{a,b,2 (0\0\0\1) 2} = {a,{a,{a,{a,...{a,{a,{a,{a,a (0\0\0\1) 2} (0\0\0\1) 2} (0\0\0\1) 2} (0\0\0\1) 2}...(0\0\0\1) 2} (0\0\0\1) 2} (0\0\0\1) 2} (0\0\0\1) 2} (b times). FGH level η0 + 1.

{a,b (0\0\0\1) 3} = {a,a (0\0\(0\0\(0\0\...0\0\(0\0\(0\0\1))))...)))) 2 (0\0\0\1) 2} (b times). FGH level (η0)2.

{a,b (0\0\0\1)(0\0\0\1) 2} = {a,a (0\0\0\1)(0\0\(0\0\(0\0\...0\0\(0\0\(0\0\1))))...)))) 2} (b times). FGH level η0^2.

{a,b (1\0\0\1) 2} = {a,a (0\0\0\1)(0\0\0\1)(0\0\0\1)...(0\0\0\1)(0\0\0\1)(0\0\0\1) 2} (b times) FGH level η0^ω.

{a,b ((0\0\0\1)\0\0\1) 2} = {a,a ((0\0\(0\0\(0\0\...0\0\(0\0\(0\0\1))))...))))\0\0\1) 2} (b times). FGH level η0^η0.

{a,b (0\1\0\1) 2} = {a,a ((((...((((0\0\0\1)\0\0\1)\0\0\1)...\0\0\1)\0\0\1)\0\0\1) 2} (b times). FGH level ε(η0 + 1).

{a,b (0\0\1\1) 2} = {a,a (0\(0\0\(0\0\(0\0\...0\0\(0\0\(0\0\1))))...))))\0\1) 2} (b times). FGH level ζ(η0 + 1).

{a,b (0\0\0\2) 2} = {a,a (0\0\0\(0\0\(0\0\(0\0\...0\0\(0\0\(0\0\1))))...)))) 2} (b times). FGH level η1 = φ(3,1).

{a,b (0\0\0\0,1) 2} = {a,a (0\0\0\b) 2}. FGH level ηω = φ(3,ω).

{a,b (0\0\0\(0\0\0\1)) 2} = {a,a (0\0\0\(0\0\(0\0\(0\0\...0\0\(0\0\(0\0\1))))...)))) 2} (b times). FGH level η(η0) = φ(3,φ(3,0)).

{a,b (0\0\0\0\1) 2} 2} = {a,a (0\0\0\(0\0\0\(0\0\0\...0\0\0\(0\0\0\(0\0\0\1))))...)))) 2} (b times). FGH level φ(4,0).

{a,b (0\0\0\0\0\1) 2}. FGH level φ(5,0).

### Beyond φ(ω,0)

#### φ(ω,0) - Γ0

(a,b (0\\1) 2} = {a,a (0\0\0\...0\0\0\1) 2} (b times). FGH level φ(ω,0).

{a,b (0\0\\1) 2}. FGH level ε(φ(ω,0) + 1).

(a,b (0\0\0\\1) 2}. FGH level ζ(φ(ω,0) + 1).

{a,b (0\0\0\0\\1) 2}. FGH level η(φ(ω,0) + 1).

{a,b (0\0\0\0\0\\1) 2}. FGH level φ(4,φ(ω,0) + 1).

{a,b (0\\2) 2}. FGH level φ(ω,1).

{a,b (0\\3) 2}. FGH level φ(ω,2).

{a,b (0\\0,1) 2} = {a,a (0\\b) 2}. FGH level φ(ω,ω).

{a,b (0\\(0\1)) 2}. FGH level φ(ω,ε0).

(a,b (0\\(0\0\1)) 2}. FGH level φ(ω,ζ0).

{a,b (0\\(0\0\0\1)) 2}. FGH level φ(ω,η0).

{a,b (0\\(0\0\0\0\1)) 2}. FGH level φ(ω,(φ(4,0)).

{a,b (0\\(0\\1)) 2}. FGH level φ(ω(φ(ω,0)).

{a,b (0\\(0\\(0\\1))) 2}. FGH level φ(ω,(φ(ω,(φ(ω,0)))).

{a,b (0\\0\1) 2}. FGH level φ(ω+1,0).

{a,b (0\\0\0\1) 2}. FGH level φ(ω+2,0).

{a,b (0\\0\\1) 2}. FGH level φ(ω2,0).

{a,b (0\\0\\0\\1) 2}. FGH level φ(ω3,0).

{a,b (0\\\1) 2} ({a,b (0 {3} 1) 2}). FGH level φ(ω^2,0).

{a,b (0 {4} 1) 2}. FGH level φ(ω^3,0).

{a,b (0 {0,1} 1) 2}. FGH level φ(ω^ω,0).

{a,b (0 {(1)1} 2}. FGH level φ(ω^ω^ω,0).

{a,b (0 {0\1} 1} 2}. FGH level φ(ε0,0).

{a,b (0 {0\0\1} 1) 2}. FGH level φ(ζ0,0).

{a,b (0 {0\0\0\1} 1) 2}. FGH level φ(η0,0).

{a,b (0 {0\\1} 1) 2}. FGH level φ(φ(ω,0),0).

{a,b (0 {0 {0\1} 1} 1) 2}. FGH level φ(φ(ε0,0),0).

{a,b (0 {0 {0\\1} 1} 1) 2}. FGH level φ(φ(φ(ω,0),0),0).

#### Γ0 - φ(1,1,0)

{a,b (0 <0/1> 1) 2} = {a,a (0 {0 {0...0 {0 {0 {2} 1} 1} 1..1} 1} 1) 2} (b times). FGH level Γ0.

{a,b (0\0 <0/1> 1) 2} = {a,a (0\0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level ε(Γ0 + 1).

{a,b (0\0\0 <0/1> 1) 2} = {a,a (0\0\0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level ζ(Γ0 + 1).

{a,b (0\0\0\0 <0/1> 1) 2} = {a,a (0\0\0\0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level η(Γ0 + 1).

{a,b (0\0\0\0\0 <0/1> 1) 2} = {a,a (0\0\0\0\0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level φ(4,Γ0 + 1).

{a,b (0\\0 <0/1> 1) 2} = {a,a (0\\0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level φ(ω,Γ0 + 1).

{a,b (0 {0\1} 0 <0/1> 1) 2} = {a,a (0 {0\1} 0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level φ(ε0,Γ0 + 1).

{a,b (0 {0 <0/1> 1} 0 <0/1> 1) 2} = {a,a (0 {0 <0/1> 1} 0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level φ(Γ0,1).

{a,b (0 {0 <0/1> 1} 1 <0/1> 1) 2} = {a,a (0 {0 <0/1> 1} 0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level φ(Γ0,2).

{a,b (0 {0 <0/1> 1} 0,1 <0/1> 1) 2} = {a,a (0 {0 <0/1> 1} 0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level φ(Γ0,ω).

{a,b (0 {0 <0/1> 1} 0 <0/1> 1 <0/1> 1) 2} = {a,a (0 {0 <0/1> 1} 0 {0 {0...0 {0 {0 {2} 1} 1} 1...1} 1} 1) 2} (b times). FGH level φ(Γ0,Γ0).