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We can put arrays inside arrays

An empty array ([]) is equal to 0

An array ending in 0 is treated like a successor

A limit array is an array not ending in 0

[0] is equal to 1 [0,0] is equal to 2 [0,0,0] is equal to 3 etc. if it is inside an array.

Make n 1

After every step increase it by one

The rules are.

  1. If there are any empty arrays, replace them with 0
  2. If the array ends with 0, cut it off
  3. If the last entry is a limit array (not ending in 0), jump inside that array and follow the rules.
  4. If the last entrfy is a successor array,
    1. call the controlling entry the last entry with a level less then the last entry.
    2. Call the copying array the part after the controlling entry. If there is no controlling entry, it is the entire array. 
    3. Replace the copying array with n+1 copies of the copying array with the final entry decremented by 1


Here's how we compare levels:

First compare the first entry of both arrays. If they are arrays, jump in them and compare them. Whatever array with the greatest first entry also has the greatest first entry is the greatesy

Repeat that but with the second, third, fourth, etc. entry, moving on to the next one if they are equal. If we reach the end of one array, the longer one is greater\

Analysis:

Levels before this are the same as regular worm notation

[[1]] has level e0

[[1],0,1] has level e0+w

[[1],1] has level w^(e0+1)

[[1],2] has level w^w^(e0+1)

[[1],[1]] has level e1

[[1],[1],[1]] has level e2

[[1,0]] has level ew

[[1,0],[1]] has level e(w+1)

[[1,0],[1],[1,0]] has level e(2w)

[[1,0],[1,0]] has level e(w^2)

[[1,0,0]] has level e(w^w)

[[1,0,0],[1,0]] has level e(w^(w+1))

[[1,0,0],[1,0],[1,0,0]] has level e(w^(2w))

[[1,0,0],[1,0,0]] has level e(w^(w^2))

[[1,0,0,0]] has level e(w^(w^w))

[[1,0,1]] has level e(e0)

[[1,0,1,0,1]] has level e(e(e0))

[[1,1]] has level z0

[[1,1],[1,1]] has level z1

[[1,1,0]] has level z(w)

[[1,1,0]] has level z(w^w)

[[1,1,0,1]] has level z(e0)

[[1,1,0,1,1]] has level z(z0)

[[1,1,1]] has level eta(0)

[[2]] has level phi(w,0)

[[2][2]] has level phi(w,1)

[[2,0]] has level phi(w,w)

[[2,0,1]] has level phi(w,e0)

[[2,1]] has level phi(w+1,0)

[[2,1,2]] has level phi(w*2,0)

[[2,2]] has level phi(w^2,0)

[[3]] has level phi(w^w,0)

[[4]] has level phi(w^w^w,0)

[[[1]]] has level phi(e0,0)

[[[[1]]]] has level phi(phi(e0,0),0)

The limit is gamma 0, not zeta 0

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