**Arrays with more than two variable, and P**

First, a short reminder of the supporting array notation given in part II:

- (i) (0,1)|x = 10^x
- (ii) (m,n+1)|x = (m,n)|(m,n)|...|(m,n)|10^frac(x) with int(x) (m,n)'s
- (iii) (m,x)|10 = (m,int(x+1))|2*5^frac(x)
- (iv) For x<2: (m+1,0)|x = (m,3)|x
- (v) For x≥2: (m+1,0)|x = (m,x)|10

(where m,n ≥ 0 are integers and x≥0 is real)

This can be easily extended to a multivariable array notation, like so:

- (i) For x≤1: (anything)|x = 10^x
- (ii) (a,b,c,...,n+1)|x = (a,b,c,...,n)|(a,b,c,...,n)|...|(a,b,c,...,n)|10^frac(x) with in(x) (a,b,c,...,n)'s
- (iii) (a,b,c,...,x)|10 = (a,b,c,...,int(x+1))|2*5^frac(x)
- (iv) For 1<x<2: (a,b,...,m+1,<k zeros>)|x = (a,b,...,m,2,<k-1 zeros>)|10^(x-1)
- (v) For x≥2: (a,b,...,m+1,<k zeros>)|x = (a,b,...,m,x,<k-1 zeros>)|10
- (vi) (a,b,...,x,<k zeros>)|10 = (a,b,...,int(x),frac(x)*10,<k-1 zeros>)|10
- (vii) (0,...,0,a,b,...,m)|x = (a,b,...,m)|x

The first 5 rules are a simple and direct extention of the 2-variable arrays notation, and Rule vii simply states that leading zeros can be ommitted.

Rule vi is an interesting one, though. It basically tells us that if we have an array which ends with (...,x,0,0,...,0) then the digits of the fractional part of x are to be distributed among the zeros. For example:

(22,7,3.14159,0,0,0,0,0)|10 = (22,7,3,1,4,1,5,9)|10.

Now, all that is left to do is to define P:

- For x<2: Px = (1,0,1)|x

- For x≥2: Px = (10^frac(x),0,...,0)|10 with int(x) zeros.

And that's it!

**P-Canonical Forms**

Just like the previous letters, any number can be written as Px (for some real number x). Here, it is actually the binary form of xPn = P(n+log x) which has the most intuitive meaning (for n≥2):

In terms of the array notation, n+1 tells us how many numbers are in the array and the digits of x tell us the what those numbers are. For example: 1.2358P4 = (1,2,3,5,8)|10

And in terms of FGH ordinals, n gives us the maximum power of ω and the digits of x give us the coefficents of the various powers of ω: 1.2358P4 ~ f_{ω⁴+ω³2+ω²3+ω5+8}(10)

(actually, these neat relations are also true for n=1 and x≥2, so 2.5P1 = (2,5)|10)

Of-course, for numbers between P2 and P10, the P-Canonical Form is also the Universal Canonical Form.

**Examples of P-Canonical Forms **

- 1 = 1P0 = P0

- 10 = 1P1 = P1

- 100 ≈ 1.0037P1 ≈ P1.0016

- 10
^{10}= F2 ≈ 1.0086P1 ≈ P1.0037

- Tritri = {3,3,3} ≈ J2.0897 ≈ 1.0210P1 ≈ P1.0090

- Graham's Number ≈ K64.492 ≈ L2.01754≈ 1.116P1 ≈ P1.0475

- {10,3,2,2} = L3 ≈ M1.4771 ≈ N1.169 = 1.169P1 ≈ P1.068

- Conway's Tetratri = 3→3→3→3 ≈ L3.011 ≈ M1.4787 ≈ N1.170 ≈ L3.011 ≈ 1.170P1 ≈ P1.068

- Conway's Tetratet = 4→4→4→4 ≈ M2.432 ≈ N1.386 = 1.386P1 ≈ P1.142

- Grand Tridecal = {10,10,10,2} = N2 = 2P1 ≈ P1.30103

- Biggol = {10,10,100,2} = M100 ≈ (2,1)|2.0037 ≈ N2.00011 = 2.00011P1 ≈ P1.30105

- N2.1 = (2,1)|10 = 2.1P1 ≈ P1.322

- Supertet = {4,4,4,4} ≈ (3,4)|3.55 ≈ N3.3356 ≈ P1.523

- N5.7 = (5,7)|10 = 5.7P1 ≈ P1.756

- General = {10,10,10,10} = N10 = P2

- Troogol = {10,10,10,100} = N100 ≈ NN1.0037 ≈ (1,0,1)|2.0016 ≈ 1.00000495P2 ≈ P2.0000022

- Fish number 1 ≈ (1,0,1)|63 ≈ 1.01P2 ≈ P2.004

- Triggol = {10,10,10,100,2} ≈ (2,0,0)|100 ≈ 2P2 ≈ P2.301

- Pentatri = {3,3,3,3,3} ≈ (2,2,3)|2.38 ≈ 2.221P2 ≈ P2.346

- 16th Goodstein number ≈ (2,2,3)|3.55 ≈ 2.226P2 ≈ P2.347

- Superpent = {5,5,5,5,5} ≈(4,4,5)|5.76 ≈ 4.447P2 ≈ P2.648

- Pentadecal = {10,10,10,10,10} = (1,0,0,0)|10 = 1P3 = P3

- 17th Goodstein number ≈ (3,3,3,4)|4.67 ≈ 3.334P3 ≈ P3.523

- Superhex = {6,6,6,6,6,6} = (5,5,5,6)|5.76 ≈5.556P3 = P3.745

- Hexadecal (?) = {10,10,10,10,10,10} = (1,0,0,0,0)|10 = 1P4 = P4

- 18th Goodstein number ≈ (5,5,5,5,5,6)|6.83 ≈ 5.556P5 ≈ P5.745

- 19th Goodstein number ≈ (7,7,7,7,7,7,7,8)|8.95 ≈ 7.778P5 ≈ P7.891

- Iteral = {10,10,10,10,10,10,10,10,10,10} = (1,0,0,0,0,0,0,0,0)|10 = 1P8 = P8

- {10,12 (1) 2} = (1,0,0,0,0,0,0,0,0,0,0)|10 = 1P10 = P10 (=Q2)

**Bonus: A Continuous Generalization of General Bowers Linear Arrays**

It turns out that for integer arguments, we get:

- {10,a,b,c,.....,n} = (n-1,...,c-1,b)|a

And replacing the 'a' by any real number x gives us a continuous version of Linear Arrays of arbitrary length. Generalizing this further to any integer base b>2 is also possible, using the usual trick of replacing all 10's with b and all 5's with b/2.

At any rate, this "bonus benifit" does not extend beyond Linear Arrays, because Pn = {10,n+2 (1) 2} rather than {10,n (1) 2}.

**The Binary Canonical Forms**

In Part I we've defined xAn as A(n+log x) for any 1<x<10 and A being either E or F or G or H.

Now we'll define Binary Forms for the other letters. Given a letter A and x<10:

- (i) If A∈{E,F,G,H,K,L,P} and x≥1 then xAn = A(n+log x)

- (ii) if A∈{J,M} and x≥2 then xAn = A(n+log
_{5}(x/2))

- (iii) For the letter N: xNn=N(n+x/10)

And the **Universal Binary Canonical Form** of a given number x is the binary counterpart of its Universal Canonical Form.

Given these definitions, we get a nice intuitive interpertation for the binary forms:

- xEn = x*10^n

- xFn = a power tower of n tens with an x on top

- mGn = 10↑↑10↑↑...10↑↑m (with n 10's)

- mHn = 10↑↑↑10↑↑↑...10↑↑↑m (with n 10's)

- mJn = 10↑↑...↑↑m with n arrows = {10,m,n}

- mKn = JJ...JJm (with n J's)

- mLn = KK...KKm (with n K's)

- mMn = (1,n)|m

- mNn = (n,m)|10

- xPn = (<the n+1 digits of x in an array>)|10

**What's Coming Next**

The system defined above is - I believe - quite intuitive. It's a nice self-contained ω^{ω}-level notation which lends itself to a simple interpertation. It also serves as a continuous extension of Bowers Linear Arrays.

So I find it very tempting to just keep it that way, and say that letter notation ends at P10. I *will* post one possible definition for Q (which is an ε₀-level function) later, but I doubt it will become part of the "official" notation.

I also plan to do two more things:

(1) Create a completely new (and much simpler) continuous notation with smoother interpolation rules. Of-course, to avoid confusion, I won't be using letters for that one.

(2) Show a variant of letter notation which allows us to: (a) order all numbers in a lexicographic order and (b) know the actual value of any given number without any calculation (for example, the number 10^10^(5.374*10^7415) will be encoded as F4-3-7415-5374)