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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.


Definition

The J Function uses algorithms defined in sandpit blogs. The sandpits help to illustrate how the J Function has evolved and where it is leading to.

The sandpits can be split into two types of functions (1) Notational and (2) Computable. The key difference is that:

(1) Notational functions generate text strings of actual FGH functions using various Veblen Hierarchy ordinals.

(2) Computable functions are based on mathematical functions that calculate explicit (large) finite numbers for various input parameters.

The J Function Sandpit \(J_8\) is the most interesting. It uses ideas from my two blogs Extended Normal Form and Unique Ordinal Representation to design an algorithm that will generate any ordinal and large number up to the Small Veblen Ordinal (SVO). Sandpit \(J_8\) is used as the basis for another function I have created, called The Alpha Function.


Comparison of the J Function Sandpits

The following is a summary of the results from the J Function sandpits to date.

Sandpit Notational or Computable Definition Growth Rate
Sandpit \(J_0\) Notational Simple description is

\(J_0(n,n^n) = f_{\epsilon_0}(n)\)

\(\epsilon_0\)
Sandpit \(J_1\) Notational Simple description is

\(J_1(2^n) = J_0(n,n^n) = f_{\epsilon_0}(n)\)

\(\epsilon_0\)
Sandpit \(J_2\) Notational Simple description is

\(J_2(n^n) = f_{\omega\uparrow\uparrow n}(n+1)\)

\(\epsilon_0\)
Sandpit \(J_3\) Notational Simple description is

\(J_3(8^n) = f_{(\omega\uparrow\uparrow(n+1))}^2(n+2)\)

\(\epsilon_0\)
Sandpit \(J_4\) Notational Rough definitions are

\(J_4(0) = f_0(2)\) then

\(J_4(n) =\) "nest the function '\(J_4(n-1)\)' to the power of 2"

\(=\) "and re-write this function in its simplest form"

\(J_4(1) = f_1(2)\)

\(J_4(2) = f_{\omega}(2)\) then

Let \(\kappa(0) = 2\) and \(\kappa(n) = \kappa(n-1)\uparrow\uparrow 2\)

\(J_4(\kappa(0)) = f_{\omega}(2)\)

\(J_4(\kappa(1)) = f_{\epsilon_0}(2)\)

\(J_4(\kappa(5)) = f_{\epsilon_{\epsilon_0}}(2) = f_{\varphi(2,0)}(2)\)

\(= f_{\zeta_0}(2) = f_{\Gamma_0}(2)\)

Function has errors which if fixed, would allow growth rate to Small Veblen Ordinal
Sandpit \(J_5\) Notational Simple Description is

\(J_5(25.00) = f_{\phi(1,0)^{\phi(1)}.\phi(1)+\phi(1,0)^{\phi(1)}.\phi(1)}^{2}(3)\)

\(= f_{\epsilon_0^{\omega}.\omega+\epsilon_0^{\omega}.\omega}^{2}(3)\)

Function has errors which if fixed, would allow growth rate to Small Veblen Ordinal
Sandpit \(J_6\) Computable \(J_6(a_1,a_2,a_3,..., a_{n-1}, a_n)\)

\(= D(a_{n[J_6(a_1,a_2,a_3,..., a_{n-1}]})\)

\(J_6(n+1,1_{[n]})\)

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

Sandpit \(J_7\) Computable \(J_7(n) = n+1 = f_0(n)\)

\(J_7(1,0) = J_7^{J_7(1)}(J_7(1))\)

\(J_7(m,n) = J_7^{J_7(m,n-1)}(m-1,J_7(m,n-1)_*)\)

and

\(J_7(k_{[n]},0_{[p]}) = \)

\(J_7^{Z}(k_{[n-2]},k_{n-1}-1,Z_*,Z_{[p]})\)

where

\(Z = J_7(k_{[n-1]},k_n-1,k_{n[p]})\)

\(J_7(n, 0_{[n]})\)

\(>> f_{\omega^n}(n) = f_{\omega^{\omega}}(n)\)

Sandpit \(J_8\) Notational Simple Description Is

\(J_8(<g>,n,p) = f_g^n(p)\)

\(= f_{(\lambda\uparrow\uparrow t)^{\gamma_e}.\gamma_c+\gamma_a}^n(p)\)

Small Veblen Ordinal
Strong D Function Computable \(D(m,n)=D(m-1,D(m,n-1))\)

and

\(D(k,m,n)\)

\(=D(k-1,D(k,m,n-1),D(k,m,n-1))\)

\(D(n,0_{[n-1]},n)\)

\(>> f_{\omega^2}(n)\)

The (old) J Function Computable \(J(k,m,n)\)

\(= D(k,0_{[m]},n)\)

and

\(J(f,g,h;k,m,n)\)

\(= J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))\)

\(J(N_{[n]})\)

\(= J(N;N;...;N)\)

\(= J(n,n,n;n,n,n;...;n,n,n)\)

\(>> f_{\omega^{\omega^2}}(n)\)


Comments and Questions

Look forward to any comments and questions. If anybody is interested, the J Function was named by my wife. The full name is the Juki Function.

Cheers B1mb0w.


References

The J Function

The following references are outdated because the J Function has since been changed. Please keep this in mind if you refer to any of these blogs.

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