Natotoin

WIP WIP WIP, yeah, so WIP I'm not even gonna say it.

"Natotoin" is the new "Notation".

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{\displaystyle Nat(a)=a\uparrow ^{f\_{\omega }(a)-2}a}

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{\displaystyle Nat(a,1)=Nat^{a}(a)}

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{\displaystyle Nat(a,b+1)=if\ f(a)=Nat(a,b)\ then\ Nat(a,b+1)=f^{a}(a)}

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{\displaystyle Nat(a,1,1)=if\ f(a)=Nat(a,a)\ then\ Nat(a,1,1)=f^{a}(a)}

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{\displaystyle Nat(a,b+1,1)=if\ f(a)=Nat(a,b,1)\ then\ Nat(a,b+1,1)=f^{a}(a)}

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{\displaystyle Nat(a,1,b+1)=if\ f(a)=Nat(a,a,b)\ then\ Nat(a,1,b+1)=f^{a}(a)}

The rules used to define Nat(a,b,1) can be used on Nat(a,b,c).

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{\displaystyle Nat(a,1,1,1)=if\ f(a)=Nat(a,a,a)\ then\ Nat(a,1,1,1)=f^{a}(a)}

etc.

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{\displaystyle Nat(a,b(1)1)=Nat(a,a,a\dots a,a,a)\ b\ times}

Nat(a,1,1(1)1) and beyond is obvious. (BEAF related but 1,2 is 1,1)

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{\displaystyle Nat(a,b(1)(1)2)=Nat(a,a(1)1(1)1(1)1\dots 1(1)1(1)1(1)1)\ b\ times}

etc.

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{\displaystyle Nat(a,b(2)1)=Nat(a,a(1)(1)(1)\dots (1)(1)(1)1)\ b\ times}

etc.

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{\displaystyle Nat(a,b(c+1)1)=Nat(a,a(c)(c)(c)\dots (c)(c)(c)1)\ b\ times}

etc.

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{\displaystyle Nat(a,b[1]1)=Nat(a,a(1(1(1\dots 1)1)1)1)\ b\ times}

etc.

Things like

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{\displaystyle Nat(a,b[1](1)1)}

are also valid.

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{\displaystyle Nat(a,b[3];[1]1)=Nat(a,a[1[1[1\dots 1]1]1]1)\ b\ times}

etc.

Continue like RIIOEL+DAN.

Time to give it a

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{\displaystyle \lfloor Boost\rceil }

!

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{\displaystyle Nat\lfloor 1\rceil (a)}

is the nesting of Nat(a,a[[a];[1]];[1]1) a times.

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{\displaystyle Nat\lfloor 1\rceil (\#)}

is also valid. (# is some sort of array)

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{\displaystyle Nat\lfloor 1\rceil \lfloor 1\rceil (a)}

works too.

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{\displaystyle \lfloor a+1\rceil \ is\ repeating\ \lfloor a\rceil 's.}

etc.

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{\displaystyle \lfloor 1,1\rceil \cdots }

is what it would normally be.

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{\displaystyle Imagine\ \lfloor \lfloor 1\rceil \rceil }

.

Nest and get

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{\displaystyle Nat(a)\&}

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Continue and finish &.

&& works too.

etc.

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{\displaystyle Nat(a)\&^{b}=Nat(a)\&\&\&\dots \&\&\&\ b\ times}

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{\displaystyle Nat(a)\&^{1,1}}

works too.

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{\displaystyle Nat(a)\&^{\lfloor 1\rceil }}

and beyond works too.

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{\displaystyle Nat(a)\&^{\&}}

works too.

etc.

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{\displaystyle Nat(a)\&\uparrow \uparrow b}

works too.

(only) On &, Nat(a) = {a,a,a} in BEAF.

Nat(&) = {&,&,&} = &{&}&

etc.

Then, get

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{\displaystyle Nat(a)Nat(\&)\&}

nest to get:

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{\displaystyle Nat(a)\&\_{1}}

etc.

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{\displaystyle Nat(a)\*\ is\ nested\ subscript\ \&'s.}

etc.

Continue...

&
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{\displaystyle {\vec {\&}}=|^{\&}(0)}