User blog:B1mb0w/J Function Sandpit J 4

J Function Sandpit \(J_4\)
The J Function is a work in progress. This sandpit defines a function called \(J_4\) which contains ideas that will be used in the final J Function. Click here for the J Function blog.

Summary
The \(J_4\) function is not really a function, but a sketch for how an ascending sequence of ordinals can be constructed beyond epsilon_0 in such a way that a function similar to \(J_3\) can be written. This eventual function (may call this \(J_5\)) will use one real number "r" as an input parameter and will effectively produce every FGH function with ordinals in the Veblen hierarchy up to SVO, i.e. a useful function to create simple references to some very large numbers which otherwise need complex notation that is difficult to read or understand.

Sign-posts or Milestones
If we focus only the FGH functions with n = 2. A relatively simple sequence of ordinals can be documented as follows. Obviously this ignores many FGH functions, but my proposal is that these Milestone ordinals can be used in a function like \(J_3\) to create every complex FGH function in-between.

Let's start with:

\(J_4(0) = 0\)

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

\(J_4(2) = f_2(2) = f_{\omega}(2)\)

\(J_4(3) = f_{\omega+1}(2)\)

\(J_4(4) = J_4(2^2) = f_{\omega+2}(2) = f_{\omega.2}(2) = f_{\omega^2}(2) = f_{\omega^{\omega}}(2) = f_{\epsilon_0}(2)\)

Without proof, the function continues quickly as follows:

\(J_4(5) = f_{\epsilon_0+1}(2)\)

\(J_4(6) = f_{\epsilon_0+2}(2) = f_{\epsilon_0+\omega}(2)\)

\(J_4(8) = f_{\epsilon_0+\omega.2}(2) = f_{\epsilon_0.2}(2) = f_{\epsilon_0.\omega}(2)\)

\(J_4(12) = f_{\epsilon_0.(\omega+1)}(2)\)

\(J_4(4^2) = f_{\epsilon_0.\epsilon_0}(2) = f_{\epsilon_0^2}(2) = f_{\epsilon_0^{\omega}}(2)\)

\(J_4(4^3) = f_{\epsilon_0^{\omega+1}}(2)\)

\(J_4(4^4) = f_{\epsilon_0^{\epsilon_0}}(2) = f_{\epsilon_1}(2)\)

Without proof, the pattern continues as follows:

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

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

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

\(J_4(\kappa(2)) = f_{\epsilon_1}(2)\)

then

\(J_4(\kappa(2)^{4^2}) = f_{\epsilon_1^{\epsilon_0^{\omega}}}(2)\)

\(J_4(\kappa(2)^{4^3}) = f_{\epsilon_1^{\epsilon_0^{\omega+1}}}(2)\)

\(J_4(\kappa(2)^{\kappa(2)}) = J_4(\kappa(3)) = f_{\epsilon_1^{\epsilon_0^{\epsilon_0}}}(2) = f_{\epsilon_1^{\epsilon_1}}(2) = f_{\epsilon_2}(2) = f_{\epsilon_{\omega}}(2)\)

Finally without proof, the pattern continues:

\(J_4(\kappa(4)) = f_{\epsilon_{\omega+1}}(2)\)

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

\(J_4(\kappa(6)) = f_{\epsilon_{\epsilon_0+1}}(2)\)

\(J_4(\kappa(7)) = f_{\epsilon_{\epsilon_0+\omega}}(2)\)

and

\(J_4(\kappa(9)) = f_{\epsilon_{\epsilon_0.\omega}}(2)\)

and

\(J_4(\kappa(13)) = f_{\epsilon_{\epsilon_0.(\omega+1)}}(2)\)

and

\(J_4(\kappa(17)) = f_{\epsilon_{\epsilon_0^{\omega}}}(2)\)

Speculation and Questions
Speculating at this point, the pattern will reach:

\(J_4(\kappa(29)) = f_{\epsilon_{\epsilon_0^{\omega+1}}}(2)\)

and

\(J_4(\kappa(41)) = f_{\epsilon_{\epsilon_1}}(2)\)

At this point, I am unsure how to proceed. I would like to get to \(\zeta_0\) ordinal for FGH with n = 2, but I do not know the correct evaluation. Is it:

\(f_{\epsilon_{\epsilon_1}}(2) = f_{\zeta_0}(2)\)

or

\(f_{\epsilon_{\epsilon_2}}(2) = f_{\zeta_0}(2)\)

or something else. Appreciate some help here. Cheers.

Note, that I have intentionally focused on n = 2 to allow this sequence to scale up to the start of the Veblen Hierarchy. As explained above my proposal is that these "Milestone" ordinals can be used in a function like \(J_3\) to create every complex FGH function in-between.

Definitions
The \(J_4\) function is not intended to be arbitrary. Here is an attempt to define the algorithm that it is based on, starting with these assumptions:

\(J_3(0) = 0\)

\(J_3(1) = f_1(2)\)

then apply a substitution rule and evaluate the resulting FGH function

\(J_3(n) =\) substitute "\(J_3(n-1)\)" for the number "2" in "\(J_3(n-1)\)"

\(= evaluate the resulting expression\)

This is not a computable function. Here are some examples of this algorithm:

\(J_3(2) = f_1(f_1(2)) = f_2(2) = f_{\omega}(2)\)

\(J_3(3) = f_{\omega}(f_{\omega}(2)) = f_{\omega+1}(2)\)

\(J_3(4) = f_{\omega+1}(f_{\omega+1}(2)) = f_{\omega^2}(2) = f_{\epsilon_0}(2)\)

and so on.

Work in progress