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. This sandpit uses the alternative rule set for the Veblen Hierarchy set out in my blog on Fundamental Sequences.
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 function (called \(J_5\)) 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:
\(J_4(0) = f_0(2)\)
\(J_4(1) = f_1(2)\)
\(J_4(2) = f_2(2) = f_{\omega}(2)\)
\(J_4(3) = f_{\omega+1}(2)\)
\(J_4(4) = J_4(2^2) = f_{\omega+2}(2) = f_{\omega.2}(2) = f_{\omega^2}(2) = f_{\omega^{\omega}}(2) = f_{\epsilon_0}(2)\)
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
\(J_4(\kappa(17)) = f_{\epsilon_{\epsilon_0^{\omega}}}(2)\)
Veblen Hierarchy[]
At this point, it is better to use Veblen Hierarchy notation. I also found the external link on the Googology Ordinal page for The Ordinal Calculator really helpful. The rule set used here is taken from my blog on Fundamental Sequences.
\(\varphi(1) = \omega\)
\(\varphi(2) = \omega^{\omega}\)
\(\varphi(\omega) = \omega\uparrow\uparrow\omega = \varphi(1,0) = \epsilon_0\)
\(\varphi(1,1) = \epsilon_1\)
and
\(\varphi(1,2) = \epsilon_2\)
Then the previous examples can be represented as follows:
\(J_4(\kappa(9)) = f_{\epsilon_{\epsilon_0.\omega}}(2) = f_{\varphi(1,\varphi(1,0).\varphi(1))}(2)\)
\(J_4(\kappa(7)) = f_{\epsilon_{\epsilon_0+\omega}}(2) = f_{\varphi(1,\varphi(1,0)+\varphi(1))}(2)\)
\(J_4(\kappa(9)) = f_{\epsilon_{\epsilon_0.\omega}}(2) = f_{\varphi(1,\varphi(1,0).\varphi(1))}(2)\)
Another change of notation is also useful. Let's define function V as follows:
\(V(\beta,p,n) = f_{\beta}^p(n)\)
Then the examples can be presented as follows:
\(J_4(\kappa(7)) = V(\varphi(1,\varphi(1,0)+\varphi(1)),1,2)\)
\(J_4(\kappa(9)) = V(\varphi(1,\varphi(1,0).\varphi(1)),1,2)\)
\(J_4(\kappa(13)) = V(\epsilon_{\epsilon_0.(\omega+1)},1,2) = V(\varphi(1,\varphi(1,0).(\varphi(1)+1)),1,2)\)
\(J_4(\kappa(17)) = V(\epsilon_{\epsilon_0^{\omega}},1,2) = V(\varphi(1,\varphi(1,0)^{\varphi(1)}),1,2)\)
Getting to \(\zeta_0\)[]
Continuing from here, I calculate the following. No proof is provided yet, but I am confident in these results:
\(J_4(\kappa(29)) = V(\epsilon_{\epsilon_0^{\omega+1}},1,2) = V(\varphi(1,\varphi(1,0)^{\varphi(1)+1}),1,2)\)
and
\(J_4(\kappa(41)) = V(\epsilon_{\epsilon_1},1,2) = V(\varphi(1,\varphi(1,1)),1,2)\)
At this point, we need to consider the \(\zeta\) function and the following results:
\(\zeta_0 = \varphi(2,0)\) from Veblen Hierarchy notation
and from my blog on Fundamental Sequences.
\(\zeta_0[2] = \varphi(2,0)[2] = \varphi_{1+1}(0)[2] = \varphi_1^2(0) = \varphi(1,\varphi(1,0)) = \epsilon_{\epsilon_0}\)
then
\(J_4(\kappa(5)) = f_{\epsilon_{\epsilon_0}}(2) = f_{\varphi(2,0)}(2) = f_{\zeta_0}(2)\)
A Reminder. I have intentionally focused on n = 2 to allow this sequence to scale into the Veblen Hierarchy. As explained above my proposal these "Milestone" ordinals will be used in a function (called \(J_5\)) to create every complex FGH function in-between.
Getting to \(\Gamma_0\)[]
Jumping ahead, the SVO is defined as the limit of the following sequence of ordinals:
\(\varphi(1, 0),\varphi(1, 0, 0),\varphi(1, 0, 0, 0),\varphi(1, 0, 0, 0, 0),\ldots\)
Therefore
\(f_{SVO}(2) = f_{\varphi(1,0,0)}(2) = f_{\Gamma_0}(2) = f_{\zeta_0}(2) = f_{\varphi(2,0)}(2) = f_{\epsilon_{\epsilon_0}}(2) = J_4(\kappa(5))\)
Definitions[]
This section will be re-written to define the \(J_4\) function using the V function and Veblen Hierarchy notation used above.
Work In Progress
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:
\(J_4(0) = f_0(2)\)
\(J_4(1) = f_1(2)\)
then apply a substitution rule and evaluate the resulting FGH function
\(J_4(n) =\) "nest the function '\(J_4(n-1)\)' to the power of 2"
\(=\) "and re-write this function in its simplest form"
This may not be a computable function. Here are some examples of this algorithm:
\(J_4(2) = f_1^2(2) = f_2(2) =\) is re-written as \(= f_{\omega}(2)\)
\(J_4(3) = f_{\omega}^2(2) =\) is re-written as \(= f_{\omega+1}(2)\)
\(J_4(4) = f_{\omega+1}^2(2) =\) is re-written as \(= f_{\omega^2}(2) = f_{\epsilon_0}(2)\)
I am confident that
\(J_4(\kappa(5)) = f_{\epsilon_{\epsilon_0}}(2) = f_{\zeta_0}(2) = f_{\Gamma_0}(2)\)
when we define:
Let \(\kappa(0) = 2\) and \(\kappa(n) = \kappa(n-1)\uparrow\uparrow 2\)
Therefore
\(\kappa(1) = 2^2 = 4\)
\(\kappa(2) = 4^4 = 256\)
\(\kappa(3) = 256^{256}\)
\(\kappa(4) = 256^{256}\uparrow\uparrow 2\)
and
\(J_4((256^{256}\uparrow\uparrow 2)\uparrow\uparrow 2) = f_{\zeta_0}(2) = f_{\Gamma_0}(2)\)