Array Omega Notation

Linear Arrays

First, some preps. In ordinal collapsing function:

• #+W is the limit ordinal for a -> #+psi(a)
• #+W_2 is the limit ordinal for a -> #+psi_1(a)
• #+W_n is the limit ordinal for a -> #+psi_(n-1)(a)

So we can define #+W_A is the limit ordinal for a -> #+psi_A(a), where A can be array space. Now for array psi notation:

• psi_(0,b,c,...)(0)[0] = 1
• psi_(a,b,c,...)(0)[0] = W_(a-1,b,c,...)+1
• psi_(a,b,c,...)(x)[0] = psi_(a,b,c,...)(x)+1
• psi_(0,...,0,b,c,...)(x)[y] with n 0s = W_(0,...,psi_(0,...,0,b,c,...)(x)[y-1] (with n 0s),b-1,c,...)(x) (with n-1 0s)
• psi_(a,b,c,...)(x)[y] = psi_(a-1,b,c,...)(x)^psi_(a,b,c,...)(x)[y-1]

Now we can define A, A can be (x1,x2,x3,x4,...) where any xn can be any ordinal, and n can be any integer.

Planer Arrays

Now we need new psi dominates all of the ordinals below than the function range.

It going to be psi_(0{1}1), and a definition is psi(0{1}1)(A,B) = W_(A,1+B).

Then, we can define psi_(x{1}1)(A,B) = W_(W_(x-1{1}1)+A,1+B) and psi_(x{1}y)(A,B) = W_(W_(x-1{1}y)+A,1+B{1}y-1)*.

* Since W_(-1{1}#) is undefined, because there is -1, it cardinal will be 0, or nothing.

Also W_(0{1}N #) have different definition.

• A+psi_(0{1}N #)(B+W_(0{1}N #)) = limit ordinal for a -> A+psi_psi_(0{1}N #)(B+W_(0{1}N #))(a)
• psi_psi_(0{1}N #)(A+W_(0{1}N #))(n) = (1+n)-th fixed point of a -> psi_(0{1}N #)(A+a)

Now we need n-row arrays on AON.

• psi_(0{1}0{1}...0{1}0{1}N #)(a,b) = W_(b{1}0{1}...0{1}0,1+a{1}N-1 #)
• psi_(a{1}0{1}...0{1}0{1}N #)(b,c) = W_(W_(a-1{1}0{1}...0{1}0{1}0{1}N #)+c{1}0{1}...0{1}0,1+b{1}N-1 #)
• So a and b need to be array, like psi_(0{1}0{1}...0{1}0{1}N #)((A),(B)) = W_(B{1}0{1}...0{1}A{1}N-1 #), where (A) and (B) are array-spaces.
• So we need another layer of psi notation, like psi_psi_#1(x,(0{1}0{1}...0{1}0{1}N #2))(a,b) = psi_#1(x,(b{1}0{1}...0{1}0,1+a{1}N-1 #2)) and psi_psi_#1((0{1}0{1}...0{1}0{1}N #2),x)(a,b) = psi_#1((b{1}0{1}...0{1}1+a{1}N-1 #2),x)
• And W_(0{1}0{1}...0{1}0{1}N #), psi_#1(x,(0{1}0{1}...0{1}0{1} N #2)), and psi_#1((0{1}0{1}...0{1}0{1} N #2),x) works like W_(0{1}N #).

Dimensional Arrays

It is easy for me to define dimensional arrays.

• psi_(0{#1}0{#2}0{#3}...0{#n}0{n}a #)(b,c) = W_(c{#1}0{#2}0{#3}...0{#n}0{n-1}1+b{n}a-1 #)
• psi_(a{#1}0{#2}0{#3}...0{#n}0{n}b #)(c,d) = W_((a-1{#1}0{#2}0{#3}...0{#n}0{n}b #)+d{#1}0{#2}0{#3}...0{#n}0{n-1}1+c{n}b-1 #)
• And W_(0{#1}0{#2}...0{#n}0{a}b #), psi_#x1(x,(0{#1}0{#2}...0{#n}0{a}b #x2)), and psi_#x1((0{#1}0{#2}...0{#n}0{a}b #x2),x) works like W_(0{1}N #).

Extended Arrays?

Yep, a new extension!

• psi_(0...0{(0,1)}N #)(A) = (1+A)-th fixed point of a -> W_(0...0{a}1{(0,1)}N-1 #)
• psi_(0...0{(0...0,a #a)}b #b)(c) = (1+A)-th fixed point of k -> W_(0...0{(0...k,a-1 #a)}1{(0...0,a #a)}b-1 #b)
• Strong Extended Arrays coming soon.

Let define #+Δ as the limit ordinal for a -> 0{#+(a)}1. By the way, psi(Δ*n) is shorthand for psi(W_(0{Δ*n}1)). The limit of current psi notation is psi(e(Δ+1)) right now.

Analysis

• W = W
• W_2 = W_2
• W_A = W_A
• psi_(0,1)(0) = psi_I(0)
• psi_(0,1)(A) = psi_I(A)
• W_(0,1) = I
• W_(W_(0,1)+A) = W_(I+A)
• psi_(1,1)(A) = psi_I_2(A)
• W_(1,1) = I_2
• W_(2,1) = I_3
• W_(A,1) = I_A
• W_(0,2) = I(1,0)
• W_(W_(0,2)+1) = W_(I(1,0)+1)
• W_(W_(0,2)+1,1) = I_(I(1,0)+1)
• W_(1,2) = I(1,1)
• W_(A,2) = I(1,A)
• W_(0,3) = I(2,0)
• W_(1,3) = I(2,1)
• W_(0,4) = I(3,0)
• W_(0,A) = I(A,0)
• W_(0,0,1) = I(1,0,0)
• W_(A,B,C,...) = I(...,C,B,A)
• psi_(0{1}1)(W_(0{1}1)^w) = chi(M^w) - Limit of normal AON
• psi_(0{1}1)(W_(0{1}1)^A) = chi(M^A)
• psi_(0{1}1)(W_(0{1}1)^W_(0{1}1)) = chi(M^M)
• W_(0{1}1) = M
• W_(W_(0{1}1)+1) = W_(M+1)
• W_(W_(0{1}1)+1,1) = I_(M+1)
• W_(W_(0{1}1)+1,A) = chi_1(A)
• W_(W_(0{1}1)+1,0,1) = chi_1(M_2)
• psi_(1{1}1)(W_(1{1}1)^w) = chi_1(M_2^w)
• W_(1{1}1) = M_2
• W_(2{1}1) = M_3
• W_(A{1}1) = M_A
• W_(0,1{1}1) = Xi(1,0)
• W_(W_(0,1{1}1)+1{1}1) = M_(Xi(1,0)+1)
• W_(1,1{1}1) = Xi(1,1)
• W_(0,2{1}1) = Xi(2,0)
• W_(A,B,C,...{1}1) = Xi(...,C,B,A)
• W_(0{1}2) = K
• W_(1{1}2) = K_2
• W_(0,1{1}2) = Sigma^1_2(1,0)
• W_(0{1}3) = Pi^1_2
• W_(0{1}A) = Pi^1_A
• W_(0{1}0,1) = Sigma^2_0(1,0)
• W_(W_(0{1}0,1)+1{1}1) = M_(Sigma^2_0(1,0)+1)
• W_(W_(0{1}0,1)+1{1}2) = K_(Sigma^2_0(1,0)+1)
• W_(W_(0{1}0,1)+1{1}A) = (Pi^1_A)_(Sigma^2_0(1,0)+1)
• W_(1{1}0,1) = Sigma^2_0(1,1)
• W_(0,1{1}0,1) = Sigma^2_0(2,0)
• W_(A,B,C,...{1}0,1) = Sigma^2_0(...,C,B,A)
• W_(0{1}1,1) = Pi^2_0
• W_(1{1}1,1) = (Pi^2_0)_2
• W_(0{1}2,1) = (Pi^2_0)^1_0
• W_(0{1}A,1) = (Pi^2_0)^1_A
• W_(0{1}0,2) = Sigma^2_1(1,0)
• W_(0{1}1,2) = Pi^2_1
• W_(0{1}2,2) = (Pi^2_1)^1_0
• W_(0{1}1,3) = Pi^2_2
• W_(0{1}0,A) = Pi^2_A
• W_(0{1}0,0,1) = Sigma^3_0(1,0)
• W_(0{1}1,0,1) = Pi^3_0
• W_(0{1}1,1,1) = (Pi^3_0)^1_0
• W_(0{1}1,0,2) = (Pi^3_0)^2_0
• W_(0{1}1,0,0,1) = Pi^4_0
• W_(0{1}0{1}1) = R in Bubby3's extension of Pi^A_B
• psi(psi_(W_(0{1}0{1}1)+1)(0)) = Limit of old psi notation
• etc.

And so on.