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My Untitled Notation (as i call it) is a new type of notation (or something like that) coined by me (alejandro magno) (yes, i am a googologist) (that means i can create my methods and my numbers). 

Definition

I will make the things simple.

It can be expressed like this: 

{a,b,c} where a is the base number (the number to be affected), c is the times number (the times the operation will be applicated to the base number), and b is the operation number (the operation to be applicated to the base number) (1 is addition, 2 is multiplication, 3 is exponentation, 4 is tetration, 5 is pentation, and so on).

Examples

{10,10,10} = 10↑↑↑↑↑↑↑↑10 aprox.

{15,8,11} = 15↑↑↑↑↑↑11 aprox.

{100,100,100} = 100↑↑↑↑↑↑↑.......↑↑↑↑↑↑↑100 with 98 arrows aprox.

Extensions

Extended Notation

This was supposed to be a system to name numbers, but it ended as an extension.

Here's how it works:

a-b-c = {a,b,c}

a-b-c-0 = a-b-c

a-b-c-1 = {{a,b,c}, {a,b,c}, {a,b,c}} (1 sublevel)

If you don't know what a sublevel is, here's a quick description:

A sublevel is an entry of my notation that is a number made with my notation (for example: {10,10,10}) that is part of another number made with my notation. We can take that number and then make the entry of another number maked with my notation be that number, and we can take that number and then make more sublevels and then more and then more and then more and then more. And so on.

a-b-c-d = (d sublevels)

And for further extensions:

a-b-c-d-0 = a-b-c-d

a-b-c-d-1 = a-b-c-(a-b-c-d) (1 sublevel)

a-b-c-d-e = (e sublevels)

...and we can continue adding and adding.

But...

there are MORE extensions to come.

For example:

a-b-c--0 = a-b-c

a-b-c--1 = a-b-c-d

a-b-c--d = a-b-c-d-d-d-...-d-d-d w/ d entries

a-b-c--d-1 = a-b-c--(a-b-c--d)

a-b-c--d-e = (e sublevels)

a-b-c--d-e-1 = a-b-c--d-(a-b-c--d-e) (1 sublevel)

a-b-c--d-e-f = (f sublevels)

a-b-c--d--e = a-b-c--d-e-e-e-...-e-e-e w/ e entries

a-b-c--d--e-f = (f sublevels)

a-b-c--d--e--f = a-b-c--d--e-f-f-f-...-f-f-f w/ f entries

a-b-c---d = a-b-c--d--d--d--...--d--d--d w/ d entries

a-b-c---d-1 = a-b-c---(a-b-c---d)

a-b-c---d-e = (e sublevels)

a-b-c---d-e-f = (f sublevels)

a-b-c---d--e = a-b-c---d-e-e-e-...-e-e-e w/ e entries

a-b-c---d--e-f = (f sublevels)

a-b-c---d--e--f = a-b-c---d--e-f-f-f-...-f-f-f w/ f entries

a-b-c---d---e = a-b-c---d--e--e--e--...--e--e--e w/ e entries

a-b-c----d = a-b-c---d---d---d---...---d---d---d w/ d entries

a-b-c-----d = a-b-c----d----d----d----...----d----d----d w/ d entries

and so on...

But...

We can continue this series.

a-b-c-^-d = a-b-c---...--- w/ d lines

a-b-c-^-0 = a-b-c

a-b-c-^-d-1 = a-b-c-^-(a-b-c-^-d)

a-b-c-^-d-e (e sublevels)

a-b-c-^-d--e = a-b-c-^-d-e-e-e-...-e-e-e w/ e entries

a-b-c-^-d--e-f = (f sublevels)

a-b-c-^-d--e--f = a-b-c-^-d--e-f-f-f-...-f-f-f w/ f entries

a-b-c-^-d---e = a-b-c-^-d--e--e--e--...--e--e--e w/ e entries

a-b-c-^-d-^-e = a-b-c-^-d---...---e w/ e lines

a-b-c-^-d-^-e-^-f = a-b-c-^-d-^-e---...---f w/ f lines

a-b-c-^--d = a-b-c-^-d-^-d-^-d-...-d-^-d-^-d w/ d entries

a-b-c-^--d-1 = a-b-c-^--(a-b-c-^--d)

a-b-c-^--d-e = (e sublevels)

a-b-c-^--d-e-f = (f sublevels)

a-b-c-^--d--e = a-b-c-^--d-e-e-e-...-e-e-e

a-b-c-^--d--e-f = (f sublevels)

a-b-c-^--d--e--f = a-b-c-^--d--e-f-f-f-...-f-f-f w/ f entries

...

a-b-c-^--d-^--e = a-b-c-^--d-^-e-^-e-^-e-^-...-^-e-^-e-^-e w/ e entries

Note: The notation has two different rules, so see my website to understand better. 

In fact, the old rule is that the last entry is the nested number and the times the nested number will be nested. The new rule says that the nester is the last entry, and the nested is the entry before the nester. And in case of only 1 entry after a-b-c, the old rule takes place.

Continuing on, we have:

a-b-c-^---d = a-b-c-^--d-^--d-^--d-^--...-^--d-^--d-^--d w/ d entries

a-b-c-^----d = a-b-c-^---d-^---d-^---d-^---...-^---d-^---d-^---d w/ d entries

and so on...

We can continue extending the system with this:

a-b-c-^-^-d = a-b-c-^---...---d w/ d lines

a-b-c-^-^--d = a-b-c-^-^-d-^-^-d-^-^-d-^-^-...-^-^-d-^-^-d-^-^-d w/ d -^-^-'s

a-b-c-^-^-^-d = a-b-c-^-^---...---d w/ d lines

and so on.

I continue expanding with:

a-b-c--^-d = a-b-c-^-^-^-...-^-^-^-d w/ d -^-'s

a-b-c--^--^-d = a-b-c--^-^-^-^-...-^-^-^-d w/ d -^-'s after --^-

a-b-c---^-d = a-b-c--^--^--^--...--^--^--^-d w/ d --^-'s

and so on...

We continue with:

a-b-c-(-^-)-^-d = a-b-c---...---^-d w/ d lines

a-b-c-(-^-)--^-d = a-b-c-(-^-)-^-^-^-...-^-^-^-d w/ d -^-'s after -(-^-)

a-b-c-(-^-)-(-^-)-^-d = a-b-c-(-^-)---...---^-d w/ d lines after -(-^-)

a-b-c-(-^-)--(-^-)-^-d = a-b-c-(-^-)-(-^-)---...---^-d w/ d lines after -(-^-)-(-^-)

a-b-c-(-^-)-^-(-^-)-^-d = a-b-c-(-^-)---...---(-^-)-^-d w/ d lines after -(-^-) and before (-^-)-^-

a-b-c-(-^-)-(-^-)-(-^-)-^-d = a-b-c-(-^-)---...---^-(-^-)-^-d w/ d lines

a-b-c-(-^-)--(-^-)-(-^-)-^-d = a-b-c-(-^-)-(-^-)-(-^-)---...---^-d w/ d lines

a-b-c-(-^-)-(-^-)--(-^-)-^-d = a-b-c-(-^-)---...---^-(-^-)-(-^-)-^-d w/ d lines

...

Like this is going out of control, i will skip some googologisms:


a-b-c-(-^-)-(-^-)-(-^-)-(-^-)-^-d = a-b-c-(-^-)-(-^-)---...---^-(-^-)-^-d w/ d lines

a-b-c-(-^-)-(-^-)-(-^-)-(-^-)-(-^-)-^-d = a-b-c-(-^-)-(-^-)-(-^-)---...---^-(-^-)-^-d w/ d lines

and so on...

Then, we define:

a-b-c-(-^--)-^-d = a-b-c-(-^-)-(-^-)-(-^-)-...-(-^-)-(-^-)-(-^-)-^-d w/ d (-^-)'s

a-b-c-(-^-^-)-^-d = a-b-c-(-^---...---)-^-d w/ d lines

...

a-b-c-((-^-)-^-)-^-d = a-b-c-(---...---^-)-^-d w/ d lines

a-b-c-(((-^-)-^-)-^-)-^-d = a-b-c-((---...---^-)-^-)-^-d w/ d lines

and so on.

We continue with:

a-b-c--(-)-^-d = a-b-c-(((...(((-^-)-^-)-^-)...)-^-)-^-)-^-)-^-d w/ d levels

a-b-c-^-(-)-^-d = a-b-c---...---(-)-^-d w/ d lines

and so on.

Then, we continue with tetration:

a-b-c-^^-d = a-b-c---...---^-(-)-^-d w/ d lines.

I would like to show you the -^^- operator possibilities, but i will not. I'm too lazy to do it. To get an definition of the idea, repeat all of the proccess we did with the -^- operator, but instead, replace all of the -^- limits with -^^-, and when -^^- is exhausted in that operator level, go to the next operator level. Then, the absolute limit of -^^- will be -^^^-, and then repeat to get -^^^^-, -^^^^^-, and so on.

We continue with the following:

a-b-c-^(n)^-d = a-b-c-^^^...^^^-d w/ n ^'s (see extended arrow notation)

So... is this the end? No! We can continue with:

a-b-c-^((n))^-d = a-b-c-^(a-b-c-^(a-b-c-^(...(a-b-c-^(a-b-c-^(a-b-c-^(n)^-d)^-d)^-d)...)^-d)^-d)^-d w/ n levels

To continue, just replace (n) to ((n)) when doing (((n))), and then continue with ((((n)))), (((((n))))), ((((((n)))))), etc.

Then, we define:

a-b-c-^(^(n)^)^d = a-b-c-^(((...(((n)))...)))^d w/ n levels

To continue, define a-b-c-^(^((n))^)^-d as a nest of this number. Continue with the others. This leads us to:

a-b-c-^(^(^(n)^)^)^-d = a-b-c-^(^(((...(((n)))...)))^)^-d w/ n levels

We can also have -^(^(^(^(n)^)^)^)^- operator, etc.

Then, we can have:

a-b-c-^(^^(n)^^)^-d = a n nest of (^()^) levels

Then, we can have (^^((n))^^), (^^(((n)))^^), etc. Then, we can have (^^(^(n)^)^^), (^^(^(^(n)^)^)^^), etc. Then, we can have (^^(^(^^(n)^^)^)^^), which is equal to a n nest of (^()^) levels after (^^()^^). We can also have (^^(^(^^(^(^^(n)^^)^)^^)^)^^), (^^(^(^^(^(^^(^(^^(n)^^)^)^^)^)^^)^)^^), etc. Then, we can define (^^(^^(n)^^)^^) as a nest of (^^(^()^)^^) operators. We can also have (^^(^^(^^(n)^^)^^)^^), (^^(^^(^^(^^(n)^^)^^)^^)^^), etc. We can define (^^^(n)^^^) as a nest of (^^()^^) levels. Then, to get to (^^^^(n)^^^^), we have to define (^(^^^()^^^)^). That is a nest of (^(^^()^^)^). Then, we can define (^^(^^^()^^^)^^) as a nest of (^^()^^). We can also get to (^^^^^()^^^^^), (^^^^^^()^^^^^^), etc. Then, we can define the (^()^()^()^) operator, which is used to nest arrows in the notation. But, things will get tedious when writing things like (^(^()^()^()^)^(^()^()^()^)^(^()^()^()^)^). If you wonder what's happenning there, is that we can apply multiple times the other operators we have seen. In fact, let's define something new:

a-b-c-^^(n)^^-d = level nest of ^()^

Or something like that. But, we can continue this notation. We can reuse our old operators, and then go into ^^^()^^^ operators, and eventually into ^()^()^()^ operators or something like that.

We can extend our system further with extensions I'm planning, but no. This is the limit of our notation.

Hope we eventually get into things like X&X&n, or even legions ({n,n/n}) or structures like X&&n.

FGH

a-b-c has level w

a-b-c-d has level w+1

a-b-c-d-e has level w+2

a-b-c--d has level w*2

a-b-c--d--e has level w*3

a-b-c---d has level w^2

a-b-c-^-d has level w^w

a-b-c-^--d has level w^(w+1)

Aarex says the following:

a-b-c--^-d has level w^(w*2)

a-b-c-(-^-)-^-d has level w^(w^2)

But i calculate:

a-b-c-^-^-d has level w^(w*2)

a-b-c-^-^--d has level w^(w*2+1)

a-b-c-^-^-^-d has level w^(w*3)

a-b-c--^-d has level w^(w^2)

a-b-c--^--^-d has level w^(w^2*2)

a-b-c---^-d has level w^(w^3)

a-b-c-(-^-)-^-d has level w^(w^w)

a-b-c-(-^-)-(-^-)-^-d has level w^(w^(w+1))

a-b-c-(-^-)--(-^-)-^-d has level w^(w^(w+2))

a-b-c-(-^-)-^-(-^-)-^-d has level w^(w^(w*2))

a-b-c-(-^-)-^-^-(-^-)-^-d has level w^(w^(w^2))

a-b-c-(-^-)--^-(-^-)-^-d has level w^(w^(w^3))

a-b-c-(-^-)---^-(-^-)-^-d has level w^(w^(w^4))

a-b-c-(-^-)-(-^-)-(-^-)-^-d has level w^(w^(w^w))

a-b-c-(-^-)--(-^-)-(-^-)-^-d has level w^(w^((w^w)+1))

a-b-c-(-^-)-^-(-^-)-(-^-)-^-d has level w^(w^((w^w)+w))

a-b-c-(-^-)-^-^-(-^-)-^-^-(-^-)-^-d has level w^(w^((w^w)+(w*2)))

a-b-c-(-^-)--^-(-^-)-(-^-)-^-d has level w^(w^((w^w)+(w^2)))

a-b-c-(-^-)-(-^-)--(-^-)-^-d has level w^(w^((w^w)*2))

a-b-c-(-^-)-(-^-)---(-^-)-^-d has level w^(w^((w^w)*3))

a-b-c-(-^-)-(-^-)-^-(-^-)-^-d has level w^(w^((w^w)^2))

a-b-c-(-^-)-(-^-)-^-^-(-^-)-^-d has level w^(w^((w^w)^2+w))

a-b-c-(-^-)-(-^-)--^-(-^-)-^-d has level w^(w^((w^w)^2*2))

More coming soon!

(Thanks for the help, Aarex!)

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