User blog:PsiCubed2/Defining Letter Notation up to N without Arrays

For reference:

Letter Notation I

Letter Notation II

Letter Notation III

The Definitions
I hereby give an alternative definition for Letter Notation up to (1,0,0,0)&#124;n (roughly P3) that does not use arrays at all:

1. The Letters available here are E,F,G,H,J,K,L,M and N.

2. A "word" is a capital letter which may be followed by one or more lower case letters. The lower case letters are limited to the same set as the upper letters (e,f,g,h,j,k,l,m,n).

(note that the capital letters themselves are - by this definition - valid words as well)

3. A valid expression is a series of concatenated words, followed by a single positive (not necessarily integer) number.

For example, we can use the words Eh, Jfl, F and Fe to create the expression:

EhJflFFe3.14

4. Repeated strings are notated by subscripts. For example:

F742 = FFFFFFF42

Fe742 = Feeeeeee42

(Fe)742 = FeFeFeFeFeFeFe42

5. To expand a given expression, we look at the word that immediately preceeds the final number. From here on we will call it "the final word".

For example, If we have MeJFe99 then "the final word" is "Fe".

6. If the "final word" is a single letter, then we expand it like so:

(a) Ex = 10^x

(b) Fx = Eex

(c) Gx = Fex

(d) Hx = Gex

(e) Jx = Gx (for x<2)

(f) Jx = Eeint(x)(2×5frac(x)) (for x≥2)

(g) Kx = Jex

(h) Lx = Kex

(i) Mx = Jjx

(j) Nx = Me (for x<2)

(k) Nx = Mjint(x-2)eint(frac(x)*10)+1(2×5frac(x)) (for x≥2)

7. If the final word has more than a single letter, then it expands like this:

(a) We'll notate the last letter of the final word by "z" and the rest of the word by "W". For example, if the final word is "Mhe" then W='Mh' and z='e'.

(b) If z='e' then:

Wex = Wex = Wint(x+1)frac(x)

For example:

Ne2.5 = N2+10.5 = N30.5 = NNN0.5 = ...

(c) otherwise, we obey the expansion rules for the capital version of z, turning any capital letters into lowercase ones. The rest of the original "final word" remains unchanged. For example:

F2.5 = Ee2.5 = E2+10.5 = E30.5 = EEE0.5 ≈ EE3.162 ≈ E1453 = 101453

So

Nf2.5 = Nee2.5 = Ne2+10.5 = Ne30.5 = NeNeNe0.5 = ...

Another example:

J2.25 = Ee2(2×50.25) ≈ Ee23 = Eee3 = EeEeEe1 = ...

So:

M2.5 = Jj2.5 = Jee2(2×50.25) = Jeee(2×50.25) ≈ Jeee3 = JeeJeeJee1 = ...

And

Nm2.5 = Njj2.5 = Njee2(2×50.25) ≈ Njeee3 = NjeeNjeeNjee1 = ...

And that's it!

Function Progression List
The progression of the functions will now be (the BEAF and up-arrow expressions are valid only for integer x>1):

Ex = 1&#124;x = 10↑x

Fx = 2&#124;x = 10↑↑x

Gx = 3&#124;x = 10↑↑↑x

Hx = 4&#124;x = 10↑↑↑↑x

Hex = 5&#124;x = 10↑5x = {10,x,5}

Hfx = 6&#124;x = 10↑6x = {10,x,6}

Hgx = 7&#124;x = 10↑7x = {10,x,7}

Hhx = 8&#124;x = 10↑8x = {10,x,8}

.

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Hh9f = 42&#124;x = 10↑42x = {10,x,42}

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Jx = (1,0)&#124;x = 10↑x10 = {10,10,x}

Kx = (1,1)&#124;x = {10,x,1,2}

Lx = (1,2)&#124;x = {10,x,2,2}

Lex = (1,3)&#124;x = {10,x,3,2}

Lfx = (1,4)&#124;x = {10,x,4,2}

Lgx = (1,5)&#124;x = {10,x,5,2}

Lhx = (1,6)&#124;x = {10,x,6,2}

Lhex = (1,7)&#124;x = {10,x,7,2}

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Lh10x = (1,42)&#124;x = {10,x,42,2}

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Mx = (2,0)&#124;x = {10,10,x,2}

Mex = (2,1)&#124;x = {10,x,1,3}

Mfx = (2,2)&#124;x = {10,x,2,3}

Mgx = (2,3)&#124;x = {10,x,3,3}

Mhx = (2,4)&#124;x = {10,x,4,3}

Mhex = (2,5)&#124;x = {10,x,5,3}

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Mjx = (3,0)&#124;x = {10,10,x,3}

Mjex = (3,1)&#124;x = {10,x,1,4}

Mjfx = (3,2)&#124;x = {10,x,2,4}

Mjgx = (3,3)&#124;x = {10,x,3,4}

Mjhx = (3,4)&#124;x = {10,x,4,4}

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Mmx = (4,0)&#124;x = {10,10,x,4}

Mmex = (4,1)&#124;x = {10,10,1,5}

Mmfx = (4,2)&#124;x = {10,10,2,5}

Mmgx = (4,3)&#124;x = {10,10,3,5}

Mmhx = (4,4)&#124;x = {10,10,4,5}

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Mmjx = (5,0)&#124;x = {10,10,x,5}

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Mmmx = (6,0)&#124;x = {10,10,x,6}

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Mmmjx = (7,0)&#124;x = {10,10,x,7}

Mm3x = (8,0)&#124;x = {10,10,x,8}

Mm3jx = (9,0)&#124;x = {10,10,x,9}

Mm4x = (10,0)&#124;x = {10,10,x,10}

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Nx = (1,0,0)&#124;x = {10,10,10,x}

Nex = (1,0,1)&#124;x = {10,x,1,1,2}

Nfx = (1,0,2)&#124;x = {10,x,2,1,2}

Ngx = (1,0,3)&#124;x = {10,x,3,1,2}

Nhx = (1,0,4)&#124;x = {10,x,4,1,2}

Nhex = (1,0,5)&#124;x = {10,x,5,1,2}

Njx = (1,1,0)&#124;x = {10,10,x,1,2}

Nkx = (1,1,1)&#124;x = {10,x,1,2,2}

Nlx = (1,1,2)&#124;x = {10,x,2,2,2}

Nlex = (1,1,3)&#124;x = {10,x,3,2,2}

Nmx = (1,2,0)&#124;x = {10,10,x,2,2}

Nmex = (1,2,1)&#124;x = {10,x,1,3,2}

Nmjx = (1,3,0)&#124;x = {10,10,x,3,2}

Nmmx = (1,4,0)&#124;x = {10,10,x,4,2}

Nmmjx = (1,5,0)&#124;x = {10,10,x,5,2}

Nnx = (2,0,0)&#124;x = {10,10,10,x,2}

Nnnx = (3,0,0)&#124;x = {10,10,10,x,3}

Nnα-1x = (α,0,0)&#124;x = {10,10,10,α,3}

And in this notation, the original P3 can be written as:

Nn910 = (10,0,0)&#124;10 = {10,10,10,10,3}

Ordinal Equivalents
As it turns out, the "lower case letters" trick is equivalent to ordinal addition. If we write down the ordinal-level of all the single letters:

E = 1

F = 2

G = 3

H = 4

J = ω

K = ω+1

L = ω+2

M = ω×2

N = ω2

Then  the ordinal strength of any "word" is simply the sum of it letters. For example:

Mjhe → ω×2+ω+4+1 → ω×3+5.

And the limit of this notation is:

Nnnnnnnnn... → ω2+ω2+ω2+... = ω3

(of-course we could add more letters for higher powers of ω, but then it won't jive with the previous version of Letter Notation because P should already be equivlaent to ωω.)