Cascading-J Notation

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IMPOSSIBLY WIP

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(
a
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b
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c
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{\displaystyle \mathrm {\AA} (a,b,c)}

= a c-ated to b

Continue like BEAF...

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# ;

Å

(
a
,
a
,
a
…
a
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a
,
a
)

{\displaystyle \mathrm {\AA} (a,b);=\mathrm {\AA} (a,a,a\dots a,a,a)}

b times

Å

(
a
,
a
)
≈

f

ω

ω

(
a
)

{\displaystyle \mathrm {\AA} (a,a)\approx f\_{\omega ^{\omega }}(a)}

etc.

Å

(
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b
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;
# 2

Å

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a
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a
…
a
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a
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a
)
;

{\displaystyle \mathrm {\AA} (a,b);2=\mathrm {\AA} (a,a,a\dots a,a,a);}

b times

etc.

Å

(
a
,
a
)
;
a
≈

f

ω

ω
1

(
a
)

{\displaystyle \mathrm {\AA} (a,a);a\approx f\_{\omega ^{\omega +1}}(a)}

Å

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b
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;
1
,
# 2

Å

(
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;
n
e
s
t

h
e
r
e

b

t
i
m
e
s

{\displaystyle \mathrm {\AA} (a,b);1,2=\mathrm {\AA} (a,a);nest\ here\ b\ times}

Å

(
a
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a
)
;
1
,
2
≈

f

ω

ω
1

1

(
a
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{\displaystyle \mathrm {\AA} (a,a);1,2\approx f\_{\omega ^{\omega +1}+1}(a)}

Å

(
a
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b
)
;
# ;

Å

(
a
,
a
)
;
a
,
a
,
a
…
a
,
a
,
a

{\displaystyle \mathrm {\AA} (a,b);;=\mathrm {\AA} (a,a);a,a,a\dots a,a,a}

b times

Å

(
a
,
a
)
;
;
≈

f

ω

ω
2

(
a
)

{\displaystyle \mathrm {\AA} (a,a);;\approx f\_{\omega ^{\omega +2}}(a)}

etc.

## Starting Js

Å

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b
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[
J
# ]

Å

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;
;
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…
;
;
;

{\displaystyle \mathrm {\AA} (a,b)[J]=\mathrm {\AA} (a,);;;\dots ;;;}

b times

Å

(
a
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a
)
[
J
]
≈

f

ω

ω
2

(
a
)

{\displaystyle \mathrm {\AA} (a,a)[J]\approx f\_{\omega ^{\omega 2}}(a)}

Å

(
a
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b
)
[
J
]
[
J
# ]

Å

(
a
,
a
)
[
J
]
;
;
;
…
;
;
;

{\displaystyle \mathrm {\AA} (a,b)[J][J]=\mathrm {\AA} (a,a)[J];;;\dots ;;;}

b times

etc.

Å

(
a
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b
)
[
J
1
# ]

Å

(
a
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a
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[
J
]
[
J
]
[
J
]
…
[
J
]
[
J
]
[
J
]

{\displaystyle \mathrm {\AA} (a,b)[J+1]=\mathrm {\AA} (a,a)[J][J][J]\dots [J][J][J]}

b times

etc.

Å

(
a
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a
)
[
J
1
]
≈

f

ω

ω
3

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a
)

{\displaystyle \mathrm {\AA} (a,a)[J+1]\approx f\_{\omega ^{\omega 3}}(a)}

Å

(
a
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b
)
[
J
c
1
# ]

Å

(
a
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a
)
[
J
c
]
[
J
c
]
[
J
c
]
…
[
J
c
]
[
J
c
]
[
J
c
]

{\displaystyle \mathrm {\AA} (a,b)[J+c+1]=\mathrm {\AA} (a,a)[J+c][J+c][J+c]\dots [J+c][J+c][J+c]}

b times

Å

(
a
,
a
)
[
J
a
]
≈

f

ω

ω

2

(
a
)

{\displaystyle \mathrm {\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

Å

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a
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a
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[
2
J
]
≈

f

ω

ω

2

ω

(
a
)

{\displaystyle \mathrm {\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...

Å

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a
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b
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[

J

2

]

{\displaystyle \mathrm {\AA} (a,b)[J\_{2}]}

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

Å

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a
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a
)
[

J

2

]
≈

f

ψ

0

(

Ω

ω

)

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a
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{\displaystyle \mathrm {\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

ψ

{\displaystyle \psi}

 is Extended Buchholz's, and the second

ψ

{\displaystyle \psi}

is PowerFlameX's to continue approximating...

Å

(
a
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a
)
[

J

a

]
≈

f

ψ

0

(

ψ

I

(

I

I

ω

)
)

(
a
)

{\displaystyle \mathrm {\AA} (a,a)[J\_{a}]\approx f\_{\psi \_{0}(\psi \_{\mathrm {I} }(\mathrm {I} ^{\mathrm {I} ^{\omega }}))}(a)}

etc.

J
(
a
,
b
# )

J

J

J

⋯

a

{\displaystyle J(a,b)=J\_{J\_{J\_{\cdots \_{a}}}}}

b times

etc.

Å

(
a
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a
)
[
J
(
a
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a
)
]
≈

f

ψ

0

(

ψ

I

(

I

I

Ω

)
)

(
a
)

{\displaystyle \mathrm {\AA} (a,a)[J(a,a)]\approx f\_{\psi \_{0}(\psi \_{\mathrm {I} }(\mathrm {I} ^{\mathrm {I} ^{\Omega }}))}(a)}

## BEYOND J

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

Å

2

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b
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{\displaystyle \mathrm {\AA} \_{2}(a,b)}

Å

2

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a
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a
)
≈

f

ψ

0

(

ψ

I

(

ε

I

1

)
)

(
a
)

{\displaystyle \mathrm {\AA} \_{2}(a,a)\approx f\_{\psi \_{0}(\psi \_{\mathrm {I} }(\varepsilon \_{\mathrm {I} +1}))}(a)}

etc.

Let's say that

ψ

I

(

I

α
1

# )

ψ

I

(

ε

I

α

1

)

{\displaystyle \psi \_{\mathrm {I} }(\mathrm {I} \_{\alpha +1})=\psi \_{\mathrm {I} }(\varepsilon \_{\mathrm {I} \_{\alpha }+1})}

Å

a

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a
,
a
)
≈

f

ψ

0

(

ψ

I

(

I

ω

)
)

(
a
)

{\displaystyle \mathrm {\AA} \_{a}(a,a)\approx f\_{\psi \_{0}(\psi \_{\mathrm {I} }(\mathrm {I} \_{\omega }))}(a)}

Basically, you do

Å

2

{\displaystyle \mathrm {\AA} \_{2}}

-arrays in the subscript to get

Å

Å

{\displaystyle \mathrm {\AA} \_{\mathrm {\AA} }}

.

etc.

## BEYOND Å

Nest the subscript to get:

B
˘

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

B
˘

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a
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a
)
≈

f

ψ

0

(

ψ

I

(

I

Ω

)
)

(
a
)

{\displaystyle {\breve {B}}(a,a)\approx f\_{\psi \_{0}(\psi \_{\mathrm {I} }(\mathrm {I} \_{\Omega }))}(a)}

etc.

## SO MUCH BEYOND

Nest the subscript there to get

C
˘

{\displaystyle {\breve {C}}}

, and you can probably see where I'm going with this...

Do the b-th layer of that to get:

§
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a
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b
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{\displaystyle \S (a,b)}

§
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a
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a
)
≈

f

ψ

0

(

ψ

I

(

I

1

(
0
)
)
)

(
a
)

{\displaystyle \S (a,a)\approx f\_{\psi \_{0}(\psi \_{\mathrm {I} }(\mathrm {I} \_{1}(0)))}(a)}

etc.