I have invented a new notation called GoogolX. It's earlier parts are on my user page, but here I'd like to post the definition up to linear arrays.

## Informal Definition

Before I formally define everything, I find that an informal definition helps.

### Definitions

"&" means any number or linear array consisting of 1 or more non-negative integers (entries) separated by commas inside a pair of square brackets.

A valid expression are two numbers which are separated by a capital X inside curly brackets which has a subscript consisting of & (definition given above).

In the expression a{X_{&}}b, "a" is the base, "b" is the prime and "&" is the dimension.

"#" means 0 or more entries inside a linear array.

"000" means 0 or more 0's in an array.

"N" means a non zero entry.

In the array [#,m,N,0,000], "m" is the nester, "N" is the conductor and "0" is the hypernester. In other words, the last nonzero entry is the conductor, the entry before it is the nester and the entry after it is the hypernester.

### Rules

Finally, the rules! Here they are.

**If the dimension is a number**

1. If the dimension is 1, then the expression is equal to the base^^^...^^base where the number of arrows is the prime, or base^^{prime}base.

2. If the prime is 1, then the expression is equal to the base to the power of itself, or a^{a}.

3. Otherwise, reduce the prime by one, nest the whole expression inside the prime and reduce the dimension by one. Order is important.

**If the dimension is an array**

4. If the conductor does not exist and there is only one entry, the array is equal to the base.

5. If the conductor exists, but the nester and the hypernester do not, nest the expression repeatedly inside the conductor with the conductor reduced by 1. The amount of nestings is equal to the expression with the conductor reduced by 1.

6. If the conductor does not exist and there is more than one entry, remove the last zero and change the second last zero to the base.

7. If the conductor and the hypernester exist, reduce the conductor by one and nest the expression repeatedly inside the hypernester. The amount of nestings is equal to the expression with the conductor reduced by one.

8. If the conductor exists and the hypernester does not, reduce the conductor by one and nest the expression repeatedly inside the hypernester. The amount of nestings is equal to the expression with the conductor reduced by one.

## Formal Definition

Yay formalism!

### Definitions

Same as above.

### Rules

1. 1. a{X_{1}}b = a^^{b}a

2. a{X_{&}}1 = a^a

3. a{X_{n+1}}b+1 = a{X_{n}}(a{X_{n+1}}b)

4. a{X_{[0]}}b = a{X_{n}}b

5. a{X_{[n+1]}}b = a{X_{a{Xa{X..[n]bb}b with a{X_{[n]}}b nestings.

6. a{X_{[000,0,0]}}b = a{X_{[#,a]}}b

7. a{X_{[#,n+1,0]}}b = a{X_{[#,n,a{X[#,n,a{X..[#,n,0]..}b}b}}b with a{X_{[#,n,a{X[#,n,0]}b]}}b nestings.

8. a{X_{[#,x,y+1]}}b = a{X_{[#,(a{X[#,(a{X[..a{X[#,x,y]}b..]}b}b}}b with a{X_{[#,x,y]}}b nestings.

## Strength

The increase in strength comes from the fact that rule 7 turned from a prime rule to a catastrophic rule, and the strongest one in my notation! I have no idea how strong the notation is now, but I'm sure it passes f_w^{3}(n), nearly sure f_w^{4}(n) and I think it's strength might even be around f_w^{w}(n), but that's probably an upper bound.

## Continuations

I have a truly miraculous, unambiguous and formal definition for separators and more, but it is too small to enter the margin of this blog post. Or the rest of it, for that matter. Next blog post coming soon!