FANDOM


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)

a$(((2)0)0)0 ≈ f_theta(Omega_theta(Omega_theta(Omega_2)))(a)

limit at f_theta(Omega_Omega)(a)

Ultimate Array Notation II & The Treasure Function

Rule 12. a$_b0 = a$_b-1((...((a)a)...)a)a

Normal Arrays has dollar level 1.

Rule 13a. treasure(0) = 1

Rule 13b. treasure(a) = a$_(treasure(a-1))a

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