Cascading-J Notation

IMPOSSIBLY WIP

Linear Arrays
$$\AA(a,b,c)$$= a c-ated to b

Continue like BEAF...

;
$$\AA(a,b); = \AA(a,a,a\dots a,a,a)$$b times

$$\AA(a,a) \approx f_{\omega^\omega}(a)$$

etc.

$$\AA(a,b);2 = \AA(a,a,a\dots a,a,a);$$b times

etc.

$$\AA(a,a);a \approx f_{\omega^{\omega+1}}(a)$$

$$\AA(a,b);1,2 = \AA(a,a);nest\ here\ b\ times$$

$$\AA(a,a);1,2 \approx f_{\omega^{\omega+1}+1}(a)$$

$$\AA(a,b);; = \AA(a,a);a,a,a\dots a,a,a$$b times

$$\AA(a,a);; \approx f_{\omega^{\omega+2}}(a)$$

etc.

Starting Js
$$\AA(a,b)[J] = \AA(a,);;;\dots;;;$$b times

$$\AA(a,a)[J] \approx f_{\omega^{\omega2}}(a)$$

$$\AA(a,b)[J][J] = \AA(a,a)[J];;;\dots;;;$$b times

etc.

$$\AA(a,b)[J+1] = \AA(a,a)[J][J][J]\dots[J][J][J]$$b times

etc.

$$\AA(a,a)[J+1] \approx f_{\omega^{\omega3}}(a)$$

$$\AA(a,b)[J+c+1] = \AA(a,a)[J+c][J+c][J+c]\dots[J+c][J+c][J+c]$$b times

$$\AA(a,a)[J+a] \approx f_{\omega^{\omega^2}}(a)$$

You can get to J+1,2, J+;, ...

Then, you get to 2J = J+J

$$\AA(a,a)[2J] \approx f_{\omega^{\omega^2+\omega}}(a)$$

etc.

etc.

Ultra-Js
You can start using this notation on Js...

Nest that to get...

$$\AA(a,b)[J_2]$$

Let's use Extended Buchholz's Function to approximate...

$$\AA(a,a)[J_2] \approx f_{\psi_0(\Omega_\omega)}(a)$$

etc.

You use the notation to its current limit and nest it to continue making this...

Let's use a mix of Extended Buchholz's and PowerFlameX's functions, where the first $$\psi$$ is Extended Buchholz's, and the second $$\psi$$is PowerFlameX's to continue approximating...

$$\AA(a,a)[J_a] \approx f_{\psi_0(\psi_\Iota(\Iota^{\Iota^\omega}))}(a)$$

etc.

$$J(a,b) = J_{J_{J_{\cdots_a}}}$$b times

etc.

$$\AA(a,a)[J(a,a)] \approx f_{\psi_0(\psi_\Iota(\Iota^{\Iota^\Omega}))}(a)$$

BEYOND J
Basically, you start doing the entire notation there over again. Nest that and get:

$$\AA_2(a,b)$$

$$\AA_2(a,a) \approx f_{\psi_0(\psi_\Iota(\varepsilon_{\Iota+1}))}(a)$$

etc.

Let's say that $$\psi_\Iota(\Iota_{\alpha+1}) = \psi_\Iota(\varepsilon_{\Iota_\alpha+1})$$

$$\AA_a(a,a) \approx f_{\psi_0(\psi_\Iota(\Iota_\omega))}(a)$$

Basically, you do $$\AA_2$$-arrays in the subscript to get $$\AA_\AA$$.

etc.

BEYOND Å
Nest the subscript to get:

$$\breve{B}(a,b)$$

$$\breve{B}(a,a) \approx f_{\psi_0(\psi_\Iota(\Iota_\Omega))}(a)$$

etc.

SO MUCH BEYOND
Nest the subscript there to get $$\breve{C}$$, and you can probably see where I'm going with this...

Do the b-th layer of that to get:

$$\S(a,b)$$

$$\S(a,a) \approx f_{\psi_0(\psi_\Iota(\Iota_1(0)))}(a)$$

etc.