User blog:B1mb0w/J Function Sandpit J 6

The J Function
The J function is a reasonably fast growing function that will be progressively modified to simplify its presentation and increase its rate of growth. This blog will explain the structure of the function and will give various calculated values. The J Function will then be used by a brand new version of the Alpha function. Click here for more information about the Alpha Function. You can also Click here to see an out-dated version of the J function.

Introduction
For an introduction, it will be useful to refer to my other blogs on the Strong D function. The following notation rules will also help to understand the behaviour of Strong D Functions and the basic structure of the J Function.

Notation:

\(D(m_{[x]}) = D(m_1,m_2,...,m_x)\)

\(D(m_{[x]},n_{[y]}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)\)

\(D(1,0_{[y]}) = D(D(1_{[y]})_{[y]})\)

Definition
The J Function is defined recursively as follows:

\(J(n) = D(n) = n+1\)

\(J(m,n) = D(n_{[J(m)]}) = D(n_{[D(m)]}) = D(n_1,n_2,...,n_{D(m)})\)

\(J(k,m,n) = D(n_{[D(m_{[D(k)]})]}) = D(n_1,n_2,...,n_{J(k,m)})\)

Basic J Function Examples:

\(J(0,n) = D(n_{[D(0)]}) = D(n_1) = D(n)\)

Because of the Leading Zero rule: L1 we get:

\(J(1,0) = J(m,0) = J(k,m,0) = D(0_{[D(m_{[D(k)]})]}) = D(0_1,0_2,...,0_{J(k,m)}) = D(0) = 1\)

\(J(1,n) = D(n_{[D(1)]}) = D(n,n) >> f_{\omega}(n-1)\)

Basic J Function Rule:

\(J(k,m,n) = D(n_{[J(k,m)]}) = J(J(k,m)-1,n) = D(n_{[D(J(k,m)-1)]}) = D(n_{[J(k,m)-1+1]}) = D(n_{[J(k,m)]})\)

Calculated Examples up to J(m,n)
\(J(1,1) = D(1,1) = 4\)

\(J(1,2) = D(2,2) = 14\)

\(J(1,4) = D(4,4) >> D(4,1) >> f_{\omega}(3)\)

\(J(2,1) = D(1_{[J(2)]}) = D(1_{[3]}) = D(1,1,1) >> D(1,0,2) >> f_{\omega+1}(3)\)

\(J(2,2) = D(2_{[J(2)]}) = D(2_{[3]}) = D(2,2,2) >> D(2,0,1) >> f_{\omega+2}(3)\)

\(J(2,3) = D(3_{[J(2)]}) = D(3_{[3]}) = D(3,3,3) >> D(3,0,1) >> f_{\omega.2}(3)\)

\(J(3,3) = D(3_{[J(3)]}) = D(3_{[4]}) = D(3,3,3,3) >> D(3,0,0,3) >> f_{\omega^2}(3)\)

and

\(J(n,n) = D(n_{[J(n)]}) = D(n_{[n+1]}) >> D(n,0_{[n-1]},n) >> f_{\omega^2}(n)\)

Calculated Examples up to J(2,m,n)
\(J(1,1,1) = D(1_{[D(1_{[D(1)]})]}) = D(1_{[D(1_{[2]})]}) = D(1_{[D(1,1)]}) = D(1_{[4]}) = D(1,1,1,1) = J(3,1)\)

or

\(J(1,1,1) = J(J(1,1)-1,1) = J(D(1_{[D(1)]})-1,1) = J(D(1_{[2]})-1,1) = J(D(1,1)-1,1)\)

\(= J(4-1,1) = J(3,1)\)

When m=0, the J function collapses to J(n) for any value of k:

\(J(k,0,n) = D(n_{[D(0_{[D(k)]})]}) = D(n_{[D(0_{[k+1]})]}) = D(n_{[D(0,0,...,0)]})\)

\(= D(n_{[1]}) = D(n) = n+1 = J(n)\)

or

\(J(k,0,n) = J(J(k,0)-1,n) = J(D(0_{[D(k)]})-1,n) = J(D(0_{[k+1]})-1,n)\)

\(= J(D(0,0,...,0)-1,n) = J(1-1,n) = J(n)\)

then

\(J(1,1,n) = J(J(1,1)-1,n) = J(D(1,1)-1,n) = J(4-1,n) = J(3,n) = D(n,n,n,n)\)

\(J(1,2,n) = J(J(1,2)-1,n) = J(D(2,2)-1,n) = J(14-1,n)\)

\(= J(13,n) = D(n,n,n,n,n,n,n,n,n,n,n,n,n,n)\)

\(J(3,0) = f_{0}(3)\)

\(J(3,0.5) = f_{0}(3)\)

\(J(3,1) = f_{1}(3)\)

\(J(3,1.5) = f_{1}^{2}(3)\)

\(J(3,2) = f_{2}(3)\)

\(J(3,2.5) = f_{2}^{2}(3)\)

\(J(3,3) = f_{\omega}(3)\)

\(J(3,3.5) = f_{\omega}^{2}(3)\)

\(J(3,4) = f_{\omega + 1}(3)\)

\(J(3,4.5) = f_{\omega + 2}(3)\)

\(J(3,5) = f_{\omega + 2}^{2}(3)\)

\(J(3,5.5) = f_{\omega^{\omega.2 + 1} + 2}^{2}(3)\)

\(J(3,6) = f_{\omega^{\omega.2 + 1}.2 + 1}(3)\)

\(J(3,6.5) = f_{\omega^{\omega.2 + 1}.2 + 2}(3)\)

\(J(3,7) = f_{\omega^{\omega.2 + 2}}^{2}(3)\)

\(J(3,7.5) = f_{\omega^{\omega.2 + 2} + 1}^{2}(3)\)

\(J(3,8) = f_{\omega^{\omega.2 + 2}.2}(3)\)

\(J(3,8.5) = f_{\omega^{\omega.2 + 2}.2 + 1}^{2}(3)\)

\(J(3,9) = f_{\omega^{\omega}}(3)\)

\(J(3,9.5) = f_{\omega^{\omega} + 1}(3)\)

\(J(3,10) = f_{\omega^{\omega} + 2}^{2}(3)\)

\(J(3,10.5) = f_{\omega^{\omega}.2}^{2}(3)\)

\(J(3,11) = f_{\omega^{\omega}.2 + 2}(3)\)

\(J(3,11.5) = f_{\omega^{\omega + 1}}^{2}(3)\)

\(J(3,12) = f_{\omega^{\omega + 1} + 1}^{2}(3)\)

\(J(3,12.5) = f_{\omega^{\omega + 1}.2}(3)\)

\(J(3,13) = f_{\omega^{\omega + 1}.2 + 1}^{2}(3)\)

\(J(3,13.5) = f_{\omega^{\omega + 2}}(3)\)

\(J(3,14) = f_{\omega^{\omega + 2} + 1}(3)\)

\(J(3,14.5) = f_{\omega^{\omega + 2} + 2}^{2}(3)\)

\(J(3,15) = f_{\omega^{\omega + 2}.2}^{2}(3)\)

\(J(3,15.5) = f_{\omega^{\omega + 2}.2 + 2}(3)\)

\(J(3,16) = f_{\omega^{\omega^{\omega.2 + 1} + 2} + 1}(3)\)

\(J(3,16.5) = f_{\omega^{\omega^{\omega.2 + 1} + 2}.2}^{2}(3)\)

\(J(3,17) = f_{\omega^{\omega^{\omega.2 + 1}.2}}^{2}(3)\)

\(J(3,17.5) = f_{\omega^{\omega^{\omega.2 + 1}.2}.2}(3)\)

\(J(3,18) = f_{\omega^{\omega^{\omega.2 + 1}.2 + 1}}(3)\)

\(J(3,18.5) = f_{\omega^{\omega^{\omega.2 + 1}.2 + 1} + 2}^{2}(3)\)

\(J(3,19) = f_{\omega^{\omega^{\omega.2 + 1}.2 + 1}.2 + 2}(3)\)

\(J(3,19.5) = f_{\omega^{\omega^{\omega.2 + 1}.2 + 2} + 2}(3)\)

\(J(3,20) = f_{\omega^{\omega^{\omega.2 + 1}.2 + 2}.2 + 1}^{2}(3)\)

\(J(3,20.5) = f_{\omega^{\omega^{\omega.2 + 2}} + 1}(3)\)

\(J(3,21) = f_{\omega^{\omega^{\omega.2 + 2}}.2 + 1}(3)\)

\(J(3,21.5) = f_{\omega^{\omega^{\omega.2 + 2} + 1}}^{2}(3)\)

\(J(3,22) = f_{\omega^{\omega^{\omega.2 + 2} + 1}.2}(3)\)

\(J(3,22.5) = f_{\omega^{\omega^{\omega.2 + 2} + 1}.2 + 2}^{2}(3)\)

\(J(3,23) = f_{\omega^{\omega^{\omega.2 + 2} + 2} + 2}^{2}(3)\)

\(J(3,23.5) = f_{\omega^{\omega^{\omega.2 + 2} + 2}.2 + 2}(3)\)

\(J(3,24) = f_{\omega^{\omega^{\omega.2 + 2}.2} + 1}^{2}(3)\)

\(J(3,24.5) = f_{\omega^{\omega^{\omega.2 + 2}.2}.2 + 1}^{2}(3)\)

\(J(3,25) = f_{\omega^{\omega^{\omega.2 + 2}.2 + 1} + 1}(3)\)

\(J(3,25.5) = f_{\omega^{\omega^{\omega.2 + 2}.2 + 1}.2 + 1}(3)\)

\(J(3,26) = f_{\omega^{\omega^{\omega.2 + 2}.2 + 2}}^{2}(3)\)

\(J(3,26.5) = f_{\omega^{\omega^{\omega.2 + 2}.2 + 2}.2}(3)\)

\(J(3,27) = f_{\omega^{\omega^{\omega}}}(3)\)

\(J(1;1,2,n) >> f_{\omega^2+\omega.2}^n(n) = f_{\omega^2+\omega.2+1}(n)\)

and

\(J(1;n,2,n) >> f_{\omega^2+\omega.2+n}(n)\) = f_{\omega^2+\omega.3}(n)\)

then

\(J(1;n,3,n) >> f_{\omega^2+\omega.4}(n)\)

WORK IN PROGRESS

\(J(1;0,0,n) >> f_{\omega^2+1}(n)\) when \(n < f_{\omega^2}(f_3(6))\)

\(J(1;1,0,0) = J(1;0,0,J(1;0,0,1)) >> f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))\)

and

\(J(1;1,0,1) = J(1;0,0,J(1;1,0,0)) >> f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6)))\)

and

\(J(1;1,0,n-1) >> f_{\omega^2+1}^n(n) = f_{\omega^2+2}(n)\) when \(n < f_{\omega^2}^2(f_3(6))\)

or

\(J(1;1,0,n) >> f_{\omega^2+2}(n)\) when \(n < f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))\)

then

\(J(1;2,0,0) = J(1;1,0,J(1;1,0,2)) >> f_{\omega^2+2}(f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6))))\)

and

\(J(1;2,0,n) >> f_{\omega^2+3}(n)\)

and

\(J(1;3,0,n) >> f_{\omega^2+4}(n)\)

then

\(J(1;n,0,n) >> f_{\omega^2+\omega}(n)\)

\(J(1;1,1,0) = J(1;J(1;1,0,1),0,J(1;1,0,1)) >> f_{\omega^2+\omega}(J(1;1,0,1))\)

\(J(1;1,1,1) = J(1;1,1,J(1;1,1,0)) >> f_{\omega^2+\omega}^2(J(1;1,0,1))\)

\(J(1;1,1,n) >> f_{\omega^2+\omega}^n(n) = f_{\omega^2+\omega+1}(n)\)

and

\(J(1;n,1,n) >> f_{\omega^2+\omega+n}(n)\) = f_{\omega^2+\omega.2}(n)\)

then

\(J(1;1,2,n) >> f_{\omega^2+\omega.2}^n(n) = f_{\omega^2+\omega.2+1}(n)\)

and

\(J(1;n,2,n) >> f_{\omega^2+\omega.2+n}(n)\) = f_{\omega^2+\omega.3}(n)\)

then

\(J(1;n,3,n) >> f_{\omega^2+\omega.4}(n)\)

and

\(J(1;n,n,n) >> f_{\omega^2+\omega.n}(n) >> f_{\omega^2.2}(n)\)

\(J(2;0,0,0) = J(1;J(1;2,2,2),J(1;2,2,2),J(1;2,2,2)) >> f_{\omega^2.2}(J(1;2,2,2))\)

\(J(2;