@ notation

***EXTREMELY WIP‼***

## Starting @s

Normal Version:

@(a) =

a

↓
a

a

{\displaystyle a\downarrow^aa}

HUGE VERSION:

@(a) =

f

α

(
a
)

{\displaystyle f\_{\alpha}(a)}

 where α is the biggest ordinal definable in Madore's Ordinal Trees using no more than a layers of branches, each branch having no more than a branches

@-(a) = @

a

{\displaystyle ^a}

(a) ≈

f

ω
1

(
a
)

{\displaystyle f\_{\omega+1}(a)}

@-(a)-b = @-

b

{\displaystyle ^b}

(a)

@-(a)-a ≈

f

ω
2

(
a
)

{\displaystyle f\_{\omega+2}(a)}

@-(a)@ = @-(a)-@(a)

@-(a)-@ = @-(a)-@-(a)

@-(a)-@ ≈

f

ω
2

(

f

ω
1

(
a
)
)

{\displaystyle f\_{\omega+2}(f\_{\omega+1}(a))}

@-(a)-@b = If ƒ(a) = @-(a)@ then @-(a)-@b =

f
b

(
a
)

{\displaystyle f^b(a)}

@-(a)-@-b = If ƒ(a) = @-(a)-@ then @-(a)-@-b =

f
b

(
a
)

{\displaystyle f^b(a)}

@-(a)-@-@ = @-(a)-@-b where b = @-(a)-@-a

@-(a)-@-@

⪆

f

ω
3

2

(
a
)

{\displaystyle \gtrapprox f\_{\omega+3}^2(a)}

etc.

## Larger @s

@--(a)-b = @-(a)-@-@-@...-@-@-@ b times

@--(a)-a ≈

f

ω
2

(
a
)

{\displaystyle f\_{\omega2}(a)}

@--(a)-@ = @--(a)-b where b = @--(a)-a

@--(a)-@ ≈

f

ω
2

2

(
a
)

{\displaystyle f\_{\omega2}^2(a)}

etc.

@--(a)--b = @--(a)-@-@-@...-@-@-@ b times

@--(a)--a ≈

f

ω
3

(
a
)

{\displaystyle f\_{\omega3}(a)}

etc.

@---(a)-b = @--(a)--a--a--a...--a--a--a b times

@---(a)-a ≈

f

ω
2

(
a
)

{\displaystyle f\_{\omega^2}(a)}

etc.

## Multi @s

@@-(a)-b = @---...---(a)-@ b times

@@-(a)-a ≈

f

ω
ω

(
a
)

{\displaystyle f\_{\omega^\omega}(a)}

etc.

@@-- and beyond will work.

@@@-(a)-b = @@---...---(a)-@

@@@-(a)-a ≈

f

ω

ω
ω

(
a
)

{\displaystyle f\_{\omega^{\omega^\omega}}(a)}

## Arrays

@-(a,2)-b = @@@...@@@-(a)-@ b times

@-(a,2)-a ≈

f

ε
0

(
a
)

{\displaystyle f\_{\varepsilon\_0}(a)}

etc.

with ,2 you can get to all the things before, including Multi @s.

@-(a,b+1)-c = @@@...@@@-(a,b)-@ c times

@-(a,a)-a ≈

f

ε
0

ω

(
a
)

{\displaystyle f\_{\varepsilon\_0\omega}(a)}

@-(a,1,2)-b is the nesting of @-(a,a)-@ b times

etc.

@-(a,1,2)-a ≈

f

ε
0

ω
1

(
a
)

{\displaystyle f\_{\varepsilon\_0\omega+1}(a)}

@-(a,1,3)-b is the nesting of @-(a,a,2)-@ b times

@-(a,1,3)-a ≈

f

ε
0

ω
2

(
a
)

{\displaystyle f\_{\varepsilon\_0\omega2}(a)}

etc.

@-(a,1,1,2)-b is the nesting of @-(a,a,a)-@ b times

etc.

@-(a,1,1,2)-a ≈

f

ε
0

ω
2

(
a
)

{\displaystyle f\_{\varepsilon\_0\omega^2}(a)}

## Brackets

@-[a]-b = @-(a,a,a...,a,a,a)-@ b times

etc.

@-[a]-a ≈

f

ε
0
2

(
a
)

{\displaystyle f\_{\varepsilon\_0^2}(a)}

## Largeness Layer Defining Notation

@-'[a]-1 = @-(a)-@

@-'[a]-2 = @-[a]-@

etc.

@-'[a]-a ≈

f

ε
0
ω

(
a
)

{\displaystyle f\_{\varepsilon\_0^\omega}(a)}

## Meta-largeness Notation

@-"[a]-b = Define all previous parts of the notation in @-'. Now, get to the b-th largeness layer for this.

@-"[a]-a ≈

f

ε
0

ε
0

ω

(
a
)

{\displaystyle f\_{\varepsilon\_0^{\varepsilon\_0^{\omega}}}(a)}

@-;[a]-1 = @-'(a)-@

@-;[a]-2 = @-"(a)-@

@-;[a]-a ≈

f

ε
1

(
a
)

{\displaystyle f\_{\varepsilon\_1}(a)}

etc.

## Meta-meta-largeness Notation

Do everything before ; in ;, and nest it b times, and you get:

@-;;[a]-b

@-;;[a]-a ≈

f

ε
1
ω

(
a
)

{\displaystyle f\_{\varepsilon\_1^\omega}(a)}

etc.

## Mata-largeness Notation

@-/[a]-b = @-;;;...;;;(a)-@

Now, do all that again to get //.

etc.

@-/[a]-a ≈

f

ε
2

(
a
)

{\displaystyle f\_{\varepsilon\_2}(a)}

@-\[a]-b = @-///...///[a]-@ b times

Nest \ to get |.

## Largerness Notation

@-`[a]-b is the b-th Largerness Layer (Largerness 1 = /, Largerness 2 = \, Largerness 3 = |, etc.)

@-`[a]-a ≈

f

ε

ω

(
a
)

{\displaystyle f\_{\varepsilon\_{\omega}}(a)}

You can get @-``[a] and beyond.

@~[a]-b = @-```...```[a]-@ b times

@~[a]-a ≈

f

ζ
0

(
a
)

{\displaystyle f\_{\zeta\_0}(a)}

You can get to @~~ and beyond.

WIP‼

## Meta-largerness Notation

@\*[a]-b = @~~~...~~~[a]-@ b times

@\*[a]-a ≈

f

η
0

(
a
)

{\displaystyle f\_{\eta\_0}(a)}

Continue...

Get to @\*\*[a]-b and so on...

## Meta-meta-largerness Notation

@&[a]-1 = @-`[a]-@

@&[a]-2 = @~[a]-@

@&[a]-3 = @\*[a]-@

etc.

@&[a]-a ≈

f

φ
(
ω
,
0
)

(
a
)

{\displaystyle f\_{\varphi(\omega,0)}(a)}

WIP‼

## Numbers

Jol = @(100)

Jool = @-(100)

Duip-jool = @-(100)-2

Triip-jool = @-(100)-3

Tetraip-jool = @-(100)-4

etc.

Jolunta = @-(100)@

Joolunta = @-(100)-@

Duip-jolunta = @-(100)-@2

Duip-joolunta = @-(100)-@-2

Triip-joolunta = @-(100)-@-3

Tetraip-joolunta = @-(100)-@-4

etc.

We can use some of Saibian's suffixes previously used in Cascading-E Notation in @ notation.

Joolda = @-(100)-@-@

Duip-joolda = @-(100)-@-@-2

etc.

Joolthra = @-(100)-@-@-@

Jooltesla = @-(100)-@-@-@-@

Joolpeta = @-(100)-@-@-@-@-@

Joolhexa = @-(100)-@-@-@-@-@-@

etc.

And now:

Dujool = @--(100)-@

Duip-dujool = @--(100)-@-2

etc.

Joolda-xata-dujool = @--(100)-@-@

Duip-joolda-xata-dujool = @--(100)-@-@-2

etc.

Joolthra-xata-dujool = @--(100)-@-@-@

etc.

Trijool = @---(100)-@

etc.

Example of things you could do:

Jool-xata-dujool-xata-trijool = @---(100)--@-@

etc.

Tetrajool = @----(100)-@

Pentajool = @-----(100)-@

Hexajool = @------(100)-@

etc.

Jooduol = @@-(100)-@

etc.

Jootriol = @@@-(100)-@

Jootetrol = @@@@-(100)-@

Joopentol = @@@@@-(100)-@

etc.

Joojudupnol = @-(100,2)-@

etc.

Another important thing using an example:

Jooduol-vasa-joojudupnol = @@-(100,2)-@

Joojutripnol = @-(100,3)-@

Joojutetrapnol = @-(100,4)-@

etc.

Jooju-unduypapnol = @-(100,1,2)-@

Recognize "unduyp"?

Joodujuol = @-[100]-@

Joodujudupnol = @-[100,2]-@

etc.

The Big = @-'[100]-@

The Monstrosity = @-"[100]-@

The Plethora = @-;[100]-@

The Insanity = @-;;[100]-@

Jooltraitha = @-;;;[100]-@

Jooltetraitha = @-;;;;[100]-@

Joolsalaitha = @-/[100]-@

Importantthingthatusesanexample:

Big-gasa-joolsalaitha = @-/'[100]-@

etc.

Joolhypersalaitha = @-\[100]-@

Joolterpersalaitha = @-|[100]-@

Joolultraitha = @-`[100]-@

Joolultraithadualna = @-``[100]-@

Joolultraithatrialna = @-```[100]-@

etc.

Joolomniaitha = @~[100]-@

Joolomniaithadualma = @~~[100]-@

Joolomniaithatrialma = @~~~[100]-@

etc.

I do intend to replace the "n" with "m" on "omniaitha".

Joolcosmolia = @\*[100]-@

Joolducosmolia = @\*\*[100]-@

Jooltricosmolia = @\*\*\*[100]-@

etc.

Joolinsania = @&[100]-@

WIP‼