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## J Function Sandpit $$J_8$$

The J Function is a work in progress. This sandpit defines a notational function called $$J_8$$ which contains ideas that will be used in the final J Function. Click here for the J Function blog.

## Summary

Edited: Late Feb 2016 to align to changes to my blog on Fundamental Sequences.

The $$J_8$$ function is another attempt to create a very fast growing function. It is an algorithm that will be defined using program code to create the notation for a large ordinal up to the size of the Small Veblen Ordinal (SVO). It uses the Unique Ordinal Representation which is defined in another blog.

## Unique Ordinal Representation

It is possible to construct a Unique Ordinal Representation of any ordinal up to SVO using a sequence of finite integers. This can be extended to construct a large number (googolism) in the form:

$$f_\gamma^n(p)$$ where $$\gamma$$ is an ordinal up to SVO, and n, p are finite integers.

To illustrate this, lets take the example from the original blog and analyse the structure of the sequence of finite integers $$<g>$$ and $$<f> = <<g>,2,4> = <1,2,0,1,0,1,1,1,0,1,0,3,0,2,2,4>$$

Sequence = $$q$$ $$<L>$$ $$t$$ $$<g_e>$$ $$<g_c>$$ $$<g_a>$$
$$<g> =$$ 1 null 2 $$<0,1>$$ $$<0,1>$$ $$<g_a>$$
$$<g_a> =$$ 1 null 1 $$<0,1>$$ $$<0,3>$$ $$<0,2>$$

and finally $$<f> = <<g>,n,p>$$ represents a function of the form $$f_\gamma^n(p)$$ where $$\gamma$$ is represented by a sequence of finite integers:

$$<g> = <1,2,0,1,0,1,1,1,0,1,0,3,0,2>$$

The sequence $$<f> = <<g>,2,4>$$ defines this large number:

$$f_{\omega^{\omega} + \omega.3 + 2}^2(4)$$

## Definition of the $$J_8$$ function

The $$J_8$$ Function uses a very recursive algorithm to define and embed sequences of the form $$<g>$$. The rules are as follows:

$$q =$$ finite integer.

$$<L> =$$ sequence of the form $$<<g_1>,<g_2>,<g_3>,...,<g_q>,p,m>$$

• where $$0 < p <= q$$ and $$0 < m$$
• but if:
• $$q = 0$$ then $$<L> =$$ null
• $$q = 1$$ then $$<L>$$ collapses to $$\omega$$

$$t =$$ finite integer but if $$q = 0$$ then $$t =$$ null

$$<g_e> =$$ sequence of the form $$<g>$$ but if $$q = 0$$ then $$<g_e> =$$ null

$$<g_c> =$$ sequence of the form $$<g>$$ but if $$q = 0$$ then $$<g_c> =$$ null.

$$<g_a> =$$ sequence of the form $$<g>$$ but if $$q = 0$$ then $$<g_a>$$ collapses to $$a$$ a finite integer.

## Some Simple Sequences

Here are some simple sequences and the number they evaluate to:

Sequence = $$q$$ $$<L>$$ $$t$$ $$<g_e>$$ $$<g_c>$$ $$<g_a>$$ $$n$$ $$p$$ = Value
$$<f> =$$ 0 null null null null 0 0 0 $$f_0^0(0) = 0$$
$$<f> =$$ 0 0 0 1 $$f_0^0(1) = 1$$
$$<f> =$$ 0 0 1 1 $$f_0^1(1) = 2 = f_{\omega}(1)$$
$$<f> =$$ 0 0 1 2 $$f_0^1(2) = 3$$
$$<f> =$$ 0 0 2 2 $$f_0^2(2) = 4 = f_1(2)$$
$$<f> =$$ 0 0 2 3 $$f_0^2(3) = 5$$
$$<f> =$$ 0 0 3 3 $$f_0^3(3) = 6 = f_1(3)$$
$$<f> =$$ 0 0 3 4 $$f_0^3(4) = 7$$
$$<f> =$$ 0 1 2 2 $$f_1^2(2) = 8 = f_2(2) = f_{\omega}(2)$$
$$<f> =$$ 0 1 2 5 $$f_1^2(5) = 20$$
$$<f> =$$ 0 1 3 3 $$f_1^3(3) = 24 = f_2(3)$$
$$<f> =$$ 0 1 3 7 $$f_1^3(7) = 56$$
$$<f> =$$ 0 1 4 4 $$f_1^4(4) = 64 = f_2(4)$$
$$<f> =$$ 0 1 4 9 $$f_1^4(9) = 144$$
$$<f> =$$ 0 1 5 5 $$f_1^5(5) = 160 = f_2(5)$$
... ... ... ... ... ... ... ... ... ...
$$<f> =$$ 0 1 9 19 $$f_1^9(19)$$
$$<f> =$$ 0 1 10 10 $$f_1^{10}(10) = f_2(10)$$
... ... ... ... ... ... ... ... ... ...
$$<f> =$$ 0 1 19 39 $$f_1^{19}(39)$$
$$<f> =$$ 0 1 20 20 $$f_1^{20}(20) = f_2(20)$$
... ... ... ... ... ... ... ... ... ...
$$<f> =$$ 0 1 23 47 $$f_1^{23}(47)$$
$$<f> =$$ 0 2 2 3 $$f_2^2(3)$$

At this point, the $$<g>$$ sequence starts to grow and many many large numbers can be generated. The general rule is illustrated as follows:

$$n <= p < f_{\gamma}(n+1) = f_{\gamma_a}(n+1)$$ therefore the range for p is uniquely defined:

$$q$$ $$<L>$$ $$t$$ $$<g_e>$$ $$<g_c>$$ $$<g_a>$$ $$n$$ $$Max(p)$$ = Value
0 0 0 $$f_0(1)-1 = 1$$ $$f_0^0(1) = 1$$
0 0 1 $$f_0(2)-1 = 2$$ $$f_0^1(2) = 3$$
0 0 2 $$f_0(3)-1 = 3$$ $$f_0^2(3) = 5$$
0 0 3 $$f_0(4)-1 = 4$$ $$f_0^3(4) = 7$$
0 1 2 $$f_1(3)-1 = 5$$ $$f_1^2(5) = 20$$
0 1 3 $$f_1(4)-1 = 7$$ $$f_1^3(7) = 56$$
0 1 4 $$f_1(5)-1 = 9$$ $$f_1^4(9) = 144$$
0 1 5 $$f_1(6)-1 = 11$$ $$f_1^5(11)$$
... ... ... ... ... ... ... ... ...
0 1 9 $$f_1(10)-1 = 19$$ $$f_1^9(19)$$
... ... ... ... ... ... ... ... ...
0 1 19 $$f_1(20)-1 = 39$$ $$f_1^{19}(39)$$
... ... ... ... ... ... ... ... ...
0 1 23 $$f_1(24)-1 = 47$$ $$f_1^{23}(47)$$
0 2 2 $$f_2(3)-1 = 23$$ $$f_2^2(23)$$

Another rule can be explained at this point to illustrate many more examples. The value of $$n$$ has a lower bound of:

$$Min(2,Max(<g>)$$

i.e. n may equal the largest finite integer in the sequence $$<g>$$ if that number is less than 2.

$$Min(2,Max(<g>) <= n <= f_{\gamma_a+1}(\gamma_a+2)-1$$ and therefore the range for n is uniquely defined:

$$q$$ $$<L>$$ $$t$$ $$<g_e>$$ $$<g_c>$$ $$<g_a>$$ $$Max(n)$$
0 0 $$f_1(2)-1 = 3$$
0 1 $$f_2(3)-1 = 23$$
0 2 $$f_3(4)-1$$
0 3 $$f_4(5)-1$$
0 4 $$f_5(6)-1$$
0 5 $$f_6(7)-1$$

## Example Sequences

The simple sequences illustrate how to construct a wide range of large numbers using different values of n and p in the sequence $$<f>$$. These examples will focus on the sequence $$<g>$$ only and will illustrate the construction of various ordinals:

Sequence = $$q$$ $$<L>$$ $$t$$ $$<g_e>$$ $$<g_c>$$ $$<g_a>$$ = Ordinal
$$<g> =$$ 0 0 $$0$$
$$<g> =$$ 0 1 1
$$<g> =$$ 0 k k
$$<g> =$$ 1 1 $$<0,1>$$ $$<0,1>$$ $$<0,0>$$ $$\omega$$
$$<g> =$$ 1 1 $$<0,1>$$ $$<0,1>$$ $$<0,1>$$ $$\omega+1$$
$$<g> =$$ 1 1 $$<0,1>$$ $$<0,1>$$ $$<0,k>$$ $$\omega+k$$
$$<g> =$$ 1 1 $$<0,1>$$ $$<0,2>$$ $$<0,k>$$ $$\omega.2+k$$
$$<g> =$$ 1 1 $$<0,1>$$ $$<0,j>$$ $$<0,k>$$ $$\omega.j+k$$
$$<g> =$$ 1 1 $$<0,2>$$ $$<0,j>$$ $$<0,k>$$ $$\omega^2.j+k$$

Continuing to modify the sequence $$<g_a>$$ will generate these ordinals:

$$<g_a>$$ = Ordinal
$$<0,k>$$ $$\omega^2.j+k$$
$$<1,1,<0,1>,<0,1>,<0,0>>$$ $$\omega^2.j+\omega$$
$$<1,1,<0,1>,<0,1>,<0,1>>$$ $$\omega^2.j+\omega+1$$
$$<1,1,<0,1>,<0,1>,<0,k>>$$ $$\omega^2.j+\omega+k$$
$$<1,1,<0,1>,<0,h>,<0,k>>$$ $$\omega^2.j+\omega.h+k$$

The sequence $$<g_a>$$ is bounded and must be less than the preceeding sequence of integers in $$<g>$$ or specifically:

$$<q,<L>,t,<g_e>>$$.

In the last entry of the above examples, we can see:

$$<1,1,<0,2>,<0,j>>$$ > $$<1,1,<0,1>,<0,h>>$$

the sequences are identical until $$2 > 1$$ in the 4th element.

More examples follow, but the sequence $$<g_a>$$ will be ignored because it is well bounded and can be derived if required.

$$q$$ $$<L>$$ $$t$$ $$<g_e>$$ $$<g_c>$$ = Ordinal
1 1 $$<0,1>$$ $$<0,1>$$ $$\omega$$
1 1 $$<0,1>$$ $$<0,2>$$ $$\omega.2$$
1 1 $$<0,1>$$ $$<0,h>$$ $$\omega.h$$
1 1 $$<0,2>$$ $$<0,h>$$ $$\omega^2.h$$
1 1 $$<0,f>$$ $$<0,h>$$ $$\omega^f.h$$
1 2 $$<0,1>$$ $$<0,j>$$ $$\omega^{\omega}.j$$
1 2 $$<0,f>$$ $$<0,j>$$ $$(\omega^{\omega})^f.j = \omega^{\omega.f}.j$$
1 2 $$<0,f>$$ $$<1,1,<0,1>,<0,1>,<0,k>>$$ $$\omega^{\omega.f}.(\omega + k)$$

## Veblen Hierarchy Sequences

To illustrate sequences using the Veblen Hierarchy function, we will assume $$<g_c> = <0,1>$$ and focus on the behavior of $$q, <L>, t, <g_e>$$ instead:

$$q$$ $$<L>$$ $$t$$ $$<g_e>$$ = Ordinal
1 1 $$<0,f>$$ $$\omega^f$$
1 2 $$<0,f>$$ $$(\omega^{\omega})^f = \omega^{\omega.f}$$
1 2 $$<1,1,<0,1>,<0,1>,<0,f>>$$ $$\omega^{\omega.(\omega + f)} = \omega^{\omega^2 + \omega.f}$$
1 2 $$<1,1,<0,1>,<0,f>,<0,0>>$$ $$\omega^{\omega.(\omega.f)} = \omega^{\omega^{f + 1}}$$
1 2 $$<1,1,<0,f>,<0,1>,<0,0>>$$ $$\omega^{\omega.(\omega^f)} = \omega^{\omega^{f + 1}}$$
1 3 $$<1,1,<0,f>,<0,1>,<0,0>>$$ $$\omega^{\omega^{\omega^{f + 1}}}$$
2 $$<<0,1>,<0,0>,1,1>$$ 1 $$<1,1,<0,1>,<0,1>,<0,0>>$$ $$(\varphi(1,0)^{\varphi(1,0)})^{\omega} = (\epsilon_0^{\epsilon_0})^{\omega}$$
2 $$<<0,1>,<0,0>,1,1>$$ 1 $$<2,<<0,1>,<0,0>>,1,$$

$$<0,1>,<0,1>,<0,0>>$$

$$(\epsilon_0^{\epsilon_0})^{\epsilon_0} = \epsilon_0^{\epsilon_0.2}$$

These examples become difficult to follow at this point. Pseudo code is explained in the next section to explain how more complex ordinals can be constructed.

## Pseudo Code Algorithm

Following is pseudo code for an algorithm that can generate these ordinals and large numbers. Lets start with a general definition of the $$J_8$$ function:

$$J_8(<g>,n,p) = f_g^n(p) = f_{(\lambda\uparrow\uparrow t)^{\gamma_e}.\gamma_c+\gamma_a}^n(p)$$

where

$$<g> == \gamma = (\lambda\uparrow\uparrow t)^{\gamma_e}.\gamma_c+\gamma_a$$

and

$$\lambda = \varphi(\gamma_{[q]})$$

The pseudo code follows this logic:

1. Sequence $$<g> = <q,<L>,t,<g_e>,<g_c>,<g_a>>$$ where q is a finite integer

2. Sequence $$<L> = <<g_1>,<g_2>,...,<g_q>,p,m>)$$

• when $$q > 1$$ then $$\lambda = \varphi^m(\gamma_{[p-1]},\gamma_p,\gamma_{[q-p]})$$ where
• $$\delta_1 < \varphi(1, 0_{[q]})$$
• WORK IN PROGRESS pseudo code for the case $$q > 1$$ will be provided
• when $$q = 1$$ then $$\lambda = \varphi(1) = \omega$$
• when $$q = 0$$ then $$\lambda =$$ null

3. t is a finite integer or null if $$q = 0$$

4. Sequence $$<g_e>$$ is recursively defined or null if $$q = 0$$ and with

• $$<0,0> < <g_e>$$ is Lower Bound
• $$<g_e> < <q,<L>,min(2,t)>$$ is Higher Bound

5. Sequence $$<g_c>$$ is recursively defined or null if $$q = 0$$ and with

• $$<0,0> < <g_c>$$ is Lower Bound
• $$<g_c> < <q,<L>,t>$$ is Higher Bound

6. Sequence $$<g_a>$$ is recursively defined or simply $$a$$ (i.e. finite integer) if $$q = 0$$ and with

• $$<g_a> < <q,<L>,t,<g_e>>$$ is Higher Bound