Natotoin

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

"Natotoin" is the new "Notation".

$$Nat(a) = a\uparrow^{f_{\omega}(a)-2}a$$

$$Nat(a,1) = Nat^a(a)$$

$$Nat(a,b+1) = if\ f(a) = Nat(a,b)\ then\ Nat(a,b+1) = f^a(a)$$

$$Nat(a,1,1) = if\ f(a) = Nat(a,a)\ then\ Nat(a,1,1) = f^a(a)$$

$$Nat(a,b+1,1) = if\ f(a) = Nat(a,b,1)\ then\ Nat(a,b+1,1) = f^a(a)$$

$$Nat(a,1,b+1) = if\ f(a) = Nat(a,a,b)\ then\ Nat(a,1,b+1) = f^a(a)$$

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

$$Nat(a,1,1,1) = if\ f(a) = Nat(a,a,a)\ then\ Nat(a,1,1,1) = f^a(a)$$

etc.

$$Nat(a,b(1)1) = Nat(a,a,a\dots a,a,a)\ b\ times$$

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

$$Nat(a,b(1)(1)2) = Nat(a,a(1)1(1)1(1)1\dots1(1)1(1)1(1)1)\ b\ times$$

etc.

$$Nat(a,b(2)1) = Nat(a,a(1)(1)(1)\dots(1)(1)(1)1)\ b\ times$$

etc.

$$Nat(a,b(c+1)1) = Nat(a,a(c)(c)(c)\dots(c)(c)(c)1)\ b\ times$$

etc.

$$Nat(a,b[1]1) = Nat(a,a(1(1(1\dots1)1)1)1)\ b\ times$$

etc.

Things like $$Nat(a,b[1](1)1)$$are also valid.

$$Nat(a,b[3];[1]1) = Nat(a,a[1[1[1\dots1]1]1]1)\ b\ times$$

etc.

Continue like RIIOEL+DAN.

Time to give it a $$\lfloor Boost\rceil$$!

$$Nat\lfloor1\rceil(a)$$is the nesting of Nat(a,aa];[1;[1]1) a times.

$$Nat\lfloor1\rceil(\#)$$is also valid. (# is some sort of array)

$$Nat\lfloor1\rceil\lfloor1\rceil(a)$$works too.

$$\lfloor a+1\rceil\ is\ repeating\ \lfloor a\rceil's.$$

etc.

$$\lfloor1,1\rceil \cdots$$is what it would normally be.

$$Imagine\ \lfloor\lfloor1\rceil\rceil$$.

Nest and get $$Nat(a)\&$$.

Continue and finish &.

&& works too.

etc.

$$Nat(a)\&^b = Nat(a)\&\&\&\dots\&\&\&\ b\ times$$

$$Nat(a)\&^{1,1}$$works too.

$$Nat(a)\&^{\lfloor1\rceil}$$and beyond works too.

$$Nat(a)\&^\&$$works too.

etc.

$$Nat(a)\&\uparrow\uparrow b$$works too.

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

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

etc.

Then, get $$Nat(a)Nat(\&)\&$$

nest to get:

$$Nat(a)\&_1$$

etc.

$$Nat(a)*\ is\ nested\ subscript\ \&'s.$$

etc.


 * [0] = &, |[1] = *, etc.

Continue...

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

Numbers
WIP