## FANDOM

10,818 Pages

### Linear Array Notation

Normal brackets have level 1

1. a$b $$\bullet$$ = (a+b)$ $$\bullet$$

2. a$$\circ$$[0 $$\bullet$$]$$\circ$$ = a$$$\circ$$a[ $$\bullet$$]$$\circ$$ 3. a$$$\circ$$[b+1$$\bullet$$]$$\circ$$ = a$$$\circ$$[b$$\bullet$$][b$$\bullet$$]...[b$$\bullet$$][b$$\bullet$$]$$\circ$$ a [b$$\bullet$$]'s Where 0 and b are the less nested numbers 4. a$[$$\diamond$$]_(b+1)$$\bullet$$ = a$[[...[[$$\diamond$$]_b ]...]_b]_b$$\bullet$$ where there are a b-brackets and the b-bracket is the bracket with the lowest level. here contains $$\diamond$$ only brackets with level bigger than b, first entry zeroes and non-first entry non-zeroes 5. a$$$\circ$$[$$\diamond$$]$$\circ$$ = a$$$\circ$$$$\diamond$$$$\diamond$$...$$\diamond$$$$\diamond$$$$\circ$$ where there are a $$\diamond$$'s ( here is 'b' 1 ) 6. a$[b$$\bullet$$]_(c+1) = a$[a$[...a$[a$[b-1$$\bullet$$]_(c+1)$$\bullet$$]_(c)$$\bullet$$...]_(c)$$\bullet$$]_(c) \

7. $$\bullet$$0 = $$\bullet$$

8. a$[$$\triangle$$,0,b$$\bullet$$] = a$[$$\triangle$$,[$$\triangle$$,0,b-1$$\bullet$$]_[$$\triangle$$,0,b-1$$\bullet$$]...[$$\triangle$$,0,b-1$$\bullet$$]_[$$\triangle$$,0,b-1$$\bullet$$],b-1$$\bullet$$]

9. a$[$$\triangle$$,c]_(b+1)$$\bullet$$ = a$[$$\triangle$$[...[$$\triangle$$[$$\triangle$$,c-1]_b ]...]_b]_b$$\bullet$$

$$\triangle$$ is an row of zeroes

### FGH

a$[0,1] ≈ f_gamma_0(a) a$[1,1] ≈ f_gamma_0+1(a)

a$[[0,1],1] ≈ f_2*gamma_0(a) a$[[1,1],1] ≈ f_w*gamma_0(a)

a$[[[1,1],1],1] ≈ f_gamma_0^w(a) a$[[[[1,1],1],1],1] ≈ f_e_(gamma_0+1)(a)

a$[[0,1]_2,1] ≈ f_phi(w,gamma_0)(a) a$[[0,1]_3,1] ≈ f_phi(w+1,gamma_0)(a)

a$[[0,1]_[0,1]_2 ,1] ≈ f_ phi(phi(w,gamma_0),gamma_0)(a) a$[[0,1]_[0,1]_[0,1]_2,1] ≈ f_phi(phi(phi(w,gamma_0),gamma_0),gamma_0)(a)

a$[0,2] ≈ f_gamma_1(a) a$[0,3] ≈ f_gamma_2(a)

a$[0,[0]] ≈ f_gamma_w(a) a$[0,[0,1]] ≈ f_gamma_gamma_0(a)

a$[0,[0,[0,1]]] ≈ f_gamma_gamma_gamma_0(a) a$[0,[0,1]_2] ≈ f_phi(1,1,0)(a)

a$[0,1[0,1]_2] ≈ f_phi(1,1,1)(a) a$[0,[[0,1]_2]] ≈ f_phi(1,2,0)(a)

a$[0,[[[0,1]_2]]] ≈ f_phi(2,0,0)(a) a$[0,[[[[0,1]_2]]]] ≈ f_phi(1,0,0,0)(a)

a$[0,[[[[[0,1]_2]]]]] ≈ f_phi(1,0,0,0,0)(a) a$[0,[1,1]_2] ≈ f_theta(Omega^omega)(a)

a$[0,[0,1]_3] ≈ f_theta(Omega^omega+1)(a) a$[0,[0,1]_[0,1]] ≈ f_theta(Omega^theta(Omega^omega))(a)

a$[0,[0,1]_[0,1]_[0,1]] ≈ f_theta(Omega^theta(Omega^theta(Omega^omega)))(a) a$[0,0,1] ≈ f_theta(Omega^Omega)(a)

a$[1,0,1] ≈ f_theta(Omega^Omega)+1(a) a$[[0,0,1],0,1] ≈ f_2*theta(Omega^Omega)(a)

a$[[0,0,1]_2,0,1] ≈ f_phi(w,theta(Omega^Omega)+1)(a) a$[[0,0,1]_[0,0,1]_2,0,1] ≈ f_phi(phi(w,theta(Omega^Omega)+1,theta(Omega^Omega)+1)(a)

a$[0,1,1] ≈ f_phi(1,0,(theta(Omega^Omega)+1))(a) a$[0,[0,0,1],1] ≈ f_phi(1,0,phi(1,0,(theta(Omega^Omega)+1)))(a)

a$[0,[0,0,1]_2,1] ≈ f_phi(1,1,(theta(Omega^Omega)+1))(a) a$[0,[[0,0,1]_2],1] ≈ f_phi(2,0,(theta(Omega^Omega)+1))(a)

a$[0,[[[0,0,1]_2]],1] ≈ f_phi(1,0,0,(theta(Omega^Omega)+1))(a) a$[0,[[[[0,0,1]_2]]],1] ≈ f_phi(1,0,0,0,(theta(Omega^Omega)+1))(a)

a$[0,0,[0,0,1]_2] ≈ f_theta((Omega^Omega)+1)(a) a$[0,0,[[0,0,1]_2]] ≈ f_theta((Omega^Omega)2)(a)

a$[0,0,[[[0,0,1]_2]]] ≈ f_theta(Omega^(Omega+1))(a) a$[0,0,[[1,0,1]_2,0,1]_2] ≈ f_theta(Omega^(Omega+theta(Omega^Omega)))(a)

a$[0,0,[0,1,1]_2] ≈ f_theta((Omega^(Omega*2))(a) a$[0,0,[0,2,1]_2] ≈ f_theta((Omega^(Omega*3))(a)

a$[0,0,[0,[0,0,1]_2,1]_2] ≈ f_theta((Omega^(Omega*theta(Omega^Omega)))(a) a$[0,0,[0,0,2]_2] ≈ f_theta((Omega^(Omega^2))(a)

a$[0,0,[0,0,1]_3] ≈ f_theta((Omega^(Omega^3))(a) a$[0,0,[0,0,1]_[0,0,1]] ≈ f_theta((Omega^(Omega^theta(Omega^Omega))))(a)

a$[0,0,[0,0,1]_[0,0,1]_[0,0,1]] ≈ f_theta((Omega^(Omega^theta(Omega^Omega^theta(Omega^Omega)))))(a) a$[0,0,0,1] ≈ f_theta(Omega^Omega^Omega)(a)

a$[0,0,0,0,1] ≈ f_theta(Omega^Omega^Omega^Omega)(a) a$[0$$\rightarrow$$1] ≈ f_theta(e_(Omega+1))(a)

### Extended Array Notation

10. This rule applies if and only if there is something in the form of a$[0$$\rightarrow_b$$c], and the 'c' is the first non-zero entry in the array. If you got somthing like 'a$[0$$\rightarrow_b$$0,c]' you have to solve 0,c first.

Rule 10: a$[0...0$$\rightarrow_b$$c$$\bullet$$] = a$[0...a$$\rightarrow_{b-1}$$a...a$$\rightarrow_{b-1}$$a$$\rightarrow_b$$c-1$$\bullet$$] where there are a new a's

### FGH

a$[0$$\rightarrow$$1] ≈ f_theta(e_(Omega+1))(a) a$[1$$\rightarrow$$1] ≈ f_theta(e_(Omega+1))+1(a)

...

a$[0$$\rightarrow$$[0$$\rightarrow$$1]_2] ≈ f_theta(e_(Omega+1)+1)(a) a$[0$$\rightarrow$$[[0$$\rightarrow$$1]_2]] ≈ f_theta(e_(Omega+1)*2)(a)

a$[0$$\rightarrow$$[[[0$$\rightarrow$$1]_2]]] ≈ f_theta(e_(Omega+1)^2)(a) a$[0$$\rightarrow$$[[[[0$$\rightarrow$$1]_2]]]] ≈ f_theta(e_(Omega+1)^Omega)(a)

a$[0$$\rightarrow$$[[[[[0$$\rightarrow$$1]_2]]]]] ≈ f_theta(e_(Omega+1)^Omega^Omega)(a) a$[0$$\rightarrow$$[[[[[[0$$\rightarrow$$1]_2]]]]] ≈ f_theta(e_(Omega+1)^Omega^Omega^Omega)(a)

a$[0$$\rightarrow$$[1$$\rightarrow$$1]_2] ≈ f_theta(e_(Omega+1)^e_(Omega+1))(a) a$[0$$\rightarrow$$[0$$\rightarrow$$1]_3] ≈ f_theta(e_(Omega+2))(a)

a$[0$$\rightarrow$$[0$$\rightarrow$$1]_4] ≈ f_theta(e_(Omega+3))(a) a$[0$$\rightarrow$$[0$$\rightarrow$$1]_[0$$\rightarrow$$1]] ≈ f_theta(e_(Omega+theta(e_(Omega+1))))(a)

a$[0$$\rightarrow$$0,1] ≈ f_theta(e_(Omega*2))(a) a$[0$$\rightarrow$$0,0,1] ≈ f_theta(e_(Omega^2))(a)

a$[0$$\rightarrow$$0,0,0,1] ≈ f_theta(e_(Omega^Omega))(a) a$[0$$\rightarrow$$0,0,0,0,1] ≈ f_theta(e_(Omega^Omega^Omega))(a)

a$[0$$\rightarrow$$0$$\rightarrow$$1] ≈ f_theta(e_(e_(Omega+1)+1))(a) a$[0$$\rightarrow$$0$$\rightarrow$$0$$\rightarrow$$1] ≈ f_theta(e_(e_(e_(Omega+1)+1)+1)))(a)

a$[0$$\rightarrow_{2}$$1] ≈ f_theta(z_(Omega+1))(a) a$[0$$\rightarrow_{3}$$1] ≈ f_theta(phi(3,Omega+1))(a)

a$[0$$\rightarrow_{4}$$1] ≈ f_theta(phi(4,Omega+1))(a) a$[0$$\rightarrow_{5}$$1] ≈ f_theta(phi(5,Omega+1))(a)

a$[0$$\rightarrow_{[0]}$$1] ≈ f_theta(phi(w,Omega+1))(a) a$[0$$\rightarrow_{[0\rightarrow_{[0]}1]}$$1] ≈ f_theta(phi(phi(w,Omega+1),Omega+1))(a)

a$[0$$\rightarrow_{[0\rightarrow_{[0\rightarrow_{[0]}1]}1]}$$1] ≈ f_theta(phi(phi(phi(w,Omega+1),Omega+1),Omega+1))(a) ### Ultimate Array Notation Rule 11. a$(b)0 = a$(b-1)[0$$\rightarrow_{(b-1)[0\rightarrow_{...(b-1)[0\rightarrow_{(b-1)[0]}1]...}1]}$$1] a nests. Normal arrays have level (1). ### FGH a$(2)0 ≈ f_theta(Omega_2)(a)

a$(3)0 ≈ f_theta(Omega_3)(a) a$(4)0 ≈ f_theta(Omega_4)(a)

a$([0])0 ≈ f_theta(Omega_w)(a) a$([0])[0$$\rightarrow$$1] ≈ f_TFB(a)

TAKEUTI FEFERMAN BUCHHOLZ ORDINAL!!!!

a$([0$$\rightarrow$$1])0 ≈ f_theta(Omega_theta(e_(Omega+1)))(a) a$((2)0)0 ≈ f_theta(Omega_theta(Omega_2))(a)