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 ~ fω⁴+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
1010 = 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+log5(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 ωω-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)

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