Now the time to extend the notation using ranking separators!

Definition

a[c X d X e X ... n]b

where all variables are nonnegative integers, and the X's are separators.

In this extension, the separators are the slash (/), and {x {X} n}, where x is an expression such that {x} is a separator, A is a separator, and n is a whole number. Comma is the shorthand for the {0} separator, and / is the shorthand for the {0}/ separator.

The number before the array is the base, the number after it is the iterator, and the separator / is called "legion separator".

Rules

• Rule 1. Base rule: a[0]b = a*b.

• Rule 2a. Entry trailing rule: a[# {X} 0]b = a[#]b ({X} is any separator).

• Rule 2b. Array trailing rule: {# {X} 0} = {# {X}} (again, {X} is any separator).

• Rule 3. Prime rule: a[%]1 = a, a[c X %]0 = 1 if c > 0, a[0 X %]0 = 0.

• Rule 4. Recursion rule: a[c#]b = a[c-1#](a[c#](b-1)) for b > 1 and c > 0.

If there are no rules above apply, start the process, which is as follows:

• Case 1. If it is a zero, skip to the next entry until the first nonzero entry.

• Case 2. If the entry is greater than 0:

  1. If it has a comma before it, change "0,n" into "b,n-1", where n is the entry and b is the iterator.

  2. If it has a separator, not comma {X} before it, and the first entry inside the separator is 0, change "{X} n" to "{X} 1 {X} n-1", and move to the first entry of the first {X} ({X} is any separator).

  3. If it has a separator, not comma {X} before it, and the first entry inside the separator is greater than 0 (X is the first entry inside the separator), replace "0 {X} n" with "0 {X-1} 0 {X-1} ... {X-1} 0 {X-1} 0 {X-1} 1 {X} n-1" where there are b zeroes between {X-1}'s (systemically, in case of {1}, change "0 {1} n" into "0,0,0,...,0,0,0,1 {1} n-1" where there are b zeroes between commas).

  4. Change the iterator into the base.

  5. The process ends.

  6. If there are legion separators (/) before it, change the {#0 / n+1%} to Sb (b is the iterator). Let S1 to be {#0 / n%} and S(n+1) = {#0 {Sn} 1 / n%}. For the simplest case, change "0 {0 / 1} 1" into "Sb", where S1 = 0,1, and S(n+1) = 0{Sn}1.

  7. If k in {k}/ is greater than 1, change "0 {k}/ m" into "0 {k-1}/ 0 {k-1}/ ... {k-1}/ 0 {k-1}/ 0 {k-1}/ 1 {k}/ m-1" with b zeroes, and do the same rules as in dimensional arrays.

  8. If there are legion separators inside {}/:

     1. For each 1 < t <= n where n is the layer of the / separator you are looking at, take At to be the separator at that layer that contains that /.

     2. Find the maximum t such that the At has no slashes, and call it a

     3. Define P and Q such that b = "p / nQ"

     4. Change A into Sb, where S1 = "P / n-1 Q" and S(n+1) = "P Sn 1 / n-1 Q".

Examples:

• 3[0{0 / 1}1]3 = 3[0{0{0,1}1}1]3

• 3[0{0 / 1}2]3 = 3[0{0{0,1}1}1{0 / 1}1]3

• 3[0{0 / 1}0{0 / 1}1]3 = 3[0{0 / 1}0{0{0,1}1}1]3

• 3[0{1 / 1}1]3 = 3[0{0 / 1}0{0 / 1}0{0{0,1}1}1]3

• 3[0{0 {0 / 1} 1 / 1}1]3 = 3[0{0{0{0,1}1}1 / 1}1]3

• 3[0{0 / 2}1]3 = 3[0{0 {0 {0 / 1} 1 / 1} 1 / 1}1]3

• 3[0{0 / 0,1}1]3 = 3[0{0 / 3}1]3 = 3[0{0 {0 {0 / 2} 1 / 2} 1 / 2}1]3

• 3[0{0 / 0 {0 / 1} 1}1]3 = 3[0{0 / 0{0{0,1}1}1}1]3

• 3[0{0 / 0 / 1}1]3 = 3[0{0 / 0 {0 / 0 {0 / 1} 1} 1}1]3

• 3[0{0 / 1 / 1}1]3 = 3[0{0 {0 {0 / 0 / 1} 1 / 0 / 1} 1 / 0 / 1}1]3

• 3[0{0 / 0 / 2}1]3 = 3[0{0 / 0 {0 / 0 {0 / 0 / 1} 1 / 1} 1 / 1}1]3

• 3[0{0 / 0 / 0,1}1]3 = 3[0{0 / 0 / 3}1]3 = 3[0{0 / 0 {0 / 0 {0 / 0 / 2} 1 / 2} 1 / 2}1]3

• 3[0{0 {1}/ 1}1]3 = 3[0{0 / 0 / 0 / 1}1]3 = 3[0{0 / 0 / 0 {0 / 0 / 0 {0 / 0 / 1} 1} 1}1]3

• 3[0{0 {0 / 1}/ 1}1]2 = 3[0{0 {0 {0 / 1} 1}/ 1}1]3 = 3[0{0 {0{0{0,1}1}1}/ 1}1]3

• 3[0{0 {0 / 1}/ 1}1]3 = 3[0{0 {0 {0 {0 {0 / 1} 1}/ 1} 1}/ 1}1]3

• 3[0{0 {0 / 1}/ 2}1]2 = 3[0{0 {0 {0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1]3

• 3[0{0 {0 / 1}/ 2}1]3 = 3[0{0 {0 {0 {0 {0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1]3

• 3[0{0 {0 / 1}/ 0,1}1]3 = 3[0{0 {0 / 1}/ 0,1}1]3

• 3[0{0 {0 / 1}/ 0 / 1}1]2 = 3[0{0 {0 / 1}/ 0 {0 {0 / 1}/ 1} 1}1]3

• 3[0{0 {0 / 1}/ 0 {0 / 1}/ 1}1]2 = 3[0{0 {0 / 1}/ 0 {0 {0 {0 / 1}/ 1} 1}/ 1}1]3

• 3[0{0 {0 / 2}/ 1}1]2 = 3[0{0 {0 {0 {0 / 1}/ 1} 1 / 1}/ 1}1]3

• 3[0{0 {0 / 0 / 1}/ 1}1]2 = 3[0{0 {0 / 0 {0 {0 / 1}/ 1} 1}/ 1}1]3

• 3[0{0 {0 {0 / 1}/ 1}/ 1}1]2 = 3[0{0 {0 {0 {0 {0 / 1}/ 1} 1}/ 1}/ 1}1]3

Explanation and analysis

Using fundamental sequences of the Buchholz's extended function forms (including extended Buchholz's simplified normal form) for the FGH ordinals.

• [0{0 / 1}1] has level ε0 = ψ0(Ω) = ψ0(ψ1(0))

• [1{0 / 1}1] has level ε0+1 = ψ0(Ω)+1 = ψ0(ψ1(0))+1

• [2{0 / 1}1] has level ε0+2 = ψ0(Ω)+2 = ψ0(ψ1(0))+2

• [0,1{0 / 1}1] has level ε0+ω = ψ0(Ω)+ω = ψ0(ψ1(0))+ψ0(1)

• [0{1}1{0 / 1}1] has level ε0+ω^ω = ψ0(Ω)+ω^ω = ψ0(ψ1(0))+ψ0(ψ0(1))

• [0{0,1}1{0 / 1}1] has level ε0+ω^ω^ω = ψ0(Ω)+ω^ω^ω = ψ0(ψ1(0))+ψ0(ψ0(ψ0(1)))

• [0{0 / 1}2] has level ε0*2 = ψ0(Ω)2 = ψ0(ψ1(0))+ψ0(ψ1(0))

• [0{0 / 1}3] has level ε0*3 = ψ0(Ω)3 = ψ0(ψ1(0))+ψ0(ψ1(0))+ψ0(ψ1(0))

• [0{0 / 1}0,1] has level ε0*ω = ω^(ε0+1) = ψ0(Ω+1) = ψ0(ψ1(0)+1)

• [0{0 / 1}0,0,1] has level ε0*ω^2 = ω^(ε0+2) = ψ0(Ω+2) = ψ0(ψ1(0)+2)

• [0{0 / 1}0{1}1] has level ε0*ω^ω = ω^(ε0+ω) = ψ0(Ω+ω) = ψ0(ψ1(0)+ψ0(1))

• [0{0 / 1}0{0,1}1] has level ε0*ω^ω^ω = ω^(ε0+ω^ω) = ψ0(Ω+ω^ω) = ψ0(ψ1(0)+ψ0(ψ0(1)))

• [0{0 / 1}0{0 / 1}1] has level ε0^2 = ω^(ε0*2) = ψ0(Ω+ψ0(Ω)) = ψ0(ψ1(0)+ψ0(ψ1(0)))

• [0{0 / 1}0{0 / 1}0{0 / 1}1] has level ε0^3 = ω^(ε0*3) = ψ0(Ω+ψ0(Ω)2) = ψ0(ψ1(0)+ψ0(ψ1(0))+ψ0(ψ1(0)))

• [0{1 / 1}1] has level ε0^ω = ω^ω^(ε0+1) = ψ0(Ω+ψ0(Ω+1)) = ψ0(ψ1(0)+ψ0(ψ1(0)+1))

• [0{2 / 1}1] has level ε0^ω^2 = ω^ω^(ε0+2) = ψ0(Ω+ψ0(Ω+2)) = ψ0(ψ1(0)+ψ0(ψ1(0)+2))

• [0{0,1 / 1}1] has level ε0^ω^ω = ω^ω^(ε0+ω) = ψ0(Ω+ψ0(Ω+ω)) = ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(1)))

• [0{0{1}1 / 1}1] has level ε0^ω^ω^ω = ω^ω^(ε0+ω^ω) = ψ0(Ω+ψ0(Ω+ω^ω)) = ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ0(1))))

• [0{0{0 / 1}1 / 1}1] has level ε0^ε0 = ω^ω^(ε0*2) = ψ0(Ω+ψ0(Ω+ψ0(Ω))) = ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ1(0))))

• [0{0{0 / 1}2 / 1}1] has level ε0^ε0^2 = ω^ω^(ε0*3) = ψ0(Ω+ψ0(Ω+ψ0(Ω)2)) = ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ1(0))+ψ0(ψ1(0))))

• [0{0{0 / 1}0,1 / 1}1] has level ε0^ε0^ω = ω^ω^ω^(ε0+1) = ψ0(Ω+ψ0(Ω+ψ0(Ω+1))) = ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ1(0)+1)))

• [0{0{0 / 1}0{0 / 1}1 / 1}1] has level ε0^ε0^ε0 = ω^ω^ω^(ε0*2) = ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω)))) = ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ1(0)))))

• [0{0{1 / 1}1 / 1}1] has level ε0^ε0^ε0^ω = ω^ω^ω^ω^(ε0+1) = ψ0(Ω+ψ0(Ω+ψ0(Ω+ψ0(Ω+1)))) = ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ1(0)+1))))

• ...

• [0{0 / 2}1] has level ε1 = ψ0(Ω2) = ψ0(ψ1(0)+ψ1(0))

• [0{0 / 2}2] has level ε1*2 = ψ0(Ω2)+ψ0(Ω2) = ψ0(ψ1(0)+ψ1(0))+ψ0(ψ1(0)+ψ1(0))

• [0{0 / 2}0,1] has level ε1*ω = ω^(ε1+1) = ψ0(Ω2+1) = ψ0(ψ1(0)+ψ1(0)+1)

• [0{0 / 2}0{0 / 1}1] has level ε1*ε0 = ω^(ε1+ε0) = ψ0(Ω2+ψ0(Ω)) = ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0)))

• [0{0 / 2}0{0 / 2}1] has level ε1^2 = ω^(ε1*2) = ψ0(Ω2+ψ0(Ω2)) = ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0)+ψ1(0)))

• [0{1 / 2}1] has level ε1^ω = ω^ω^(ε1+1) = ψ0(Ω2+ψ0(Ω2+1)) = ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0)+ψ1(0)+1))

• [0{0 {0 / 1} 1 / 2}1] has level ε1^ε0 = ω^ω^(ε1+ε0) = ψ0(Ω2+ψ0(Ω2+ψ0(Ω))) = ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0))))

• [0{0 {0 / 2} 1 / 2}1] has level ε1^ε1 = ω^ω^(ε1*2) = ψ0(Ω2+ψ0(Ω2+ψ0(Ω2))) = ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0)+ψ1(0))))

• [0{0 {0 / 2} 0,1 / 2}1] has level ε1^ε1^ω = ω^ω^ω^(ε1+1) = ψ0(Ω2+ψ0(Ω2+ψ0(Ω2+1))) = ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0)+ψ1(0)+ψ0(ψ1(0)+ψ1(0)+1)))

• [0{0 / 3}1] has level ε2 = ψ0(Ω3) = ψ0(ψ1(0)+ψ1(0)+ψ1(0))

• [0{0 / 3}0,1] has level ε2*ω = ω^(ε2+1) = ψ0(Ω3+1) = ψ0(ψ1(0)+ψ1(0)+ψ1(0)+1)

• [0{0 / 4}1] has level ε3 = ψ0(Ω4) = ψ0(ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0))

• [0{0 / 5}1] has level ε4 = ψ0(Ω5) = ψ0(ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0))

• ...

• [0{0 / 0,1}1] has level εω = ψ0(Ωω) = ψ0(ψ1(1))

• [0{0 / 0,1}0,1] has level εω*ω = ω^(εω+1) = ψ0(Ωω+1) = ψ0(ψ1(1)+1)

• [0{0 / 0,1}0{0 / 0,1}1] has level εω^2 = ω^(εω*2) = ψ0(Ωω+ψ0(Ωω)) = ψ0(ψ1(1)+ψ0(ψ1(1)))

• [0{1 / 0,1}1] has level εω^ω = ω^ω^(εω+1) = ψ0(Ωω+ψ0(Ωω+1)) = ψ0(ψ1(1)+ψ0(ψ1(1)+1))

• [0{0 {0 / 0,1} 1 / 0,1}1] has level εω^εω = ω^ω^(εω*2) = ψ0(Ωω+ψ0(Ωω+ψ0(Ωω))) = ψ0(ψ1(1)+ψ0(ψ1(1)+ψ0(ψ1(1))))

• [0{0 {0 / 0,1} 0,1 / 0,1}1] has level εω^εω^ω = ω^ω^ω^(εω+1) = ψ0(Ωω+ψ0(Ωω+ψ0(Ωω+1))) = ψ0(ψ1(1)+ψ0(ψ1(1)+ψ0(ψ1(1)+1)))

• [0{0 / 1,1}1] has level ε(ω+1) = ψ0(Ωω+Ω) = ψ0(ψ1(1)+ψ1(0))

• [0{0 / 2,1}1] has level ε(ω+2) = ψ0(Ωω+Ω2) = ψ0(ψ1(1)+ψ1(0)+ψ1(0))

• [0{0 / 0,2}1] has level ε(ω2) = ψ0(Ωω2) = ψ0(ψ1(1)+ψ1(1))

• [0{0 / 0,3}1] has level ε(ω3) = ψ0(Ωω3) = ψ0(ψ1(1)+ψ1(1)+ψ1(1))

• [0{0 / 0,0,1}1] has level ε(ω^2) = ψ0(Ωω^2) = ψ0(ψ1(2))

• [0{0 / 0,0,0,1}1] has level ε(ω^3) = ψ0(Ωω^3) = ψ0(ψ1(3))

• [0{0 / 0{1}1}1] has level ε(ω^ω) = ψ0(Ωω^ω) = ψ0(ψ1(ψ0(1)))

• [0{0 / 0{2}1}1] has level ε(ω^ω^2) = ψ0(Ωω^ω^2) = ψ0(ψ1(ψ0(2)))

• [0{0 / 0{0,1}1}1] has level ε(ω^ω^ω) = ψ0(Ωω^ω^ω) = ψ0(ψ1(ψ0(ψ0(1))))

• [0{0 / 0{0{1}1}1}1] has level ε(ω^ω^ω^ω) = ψ0(Ωω^ω^ω^ω) = ψ0(ψ1(ψ0(ψ0(ψ0(1)))))

• ...

• [0{0 / 0{0 / 1}1}1] has level ε(ε0) = ψ0(Ωψ0(Ω)) = ψ0(ψ1(ψ0(ψ1(0))))

• [0{0 / 0{0 / 1}2}1] has level ε(ε0*2) = ψ0(Ωψ0(Ω)2) = ψ0(ψ1(ψ0(ψ1(0))+ψ0(ψ1(0))))

• [0{0 / 0{0 / 1}0,1}1] has level ε(ε0*ω) = ψ0(Ωψ0(Ω+1)) = ψ0(ψ1(ψ0(ψ1(0)))+ψ1(ψ0(ψ1(0)+1)))

• [0{0 / 0{0 / 1}0{0 / 1}1}1] has level ε(ε0^2) = ψ0(Ωψ0(Ω+ψ0(Ω))) = ψ0(ψ1(ψ0(ψ1(0)))+ψ1(ψ0(ψ1(0)+ψ0(ψ1(0)))))

• [0{0 / 0{1 / 1}1}1] has level ε(ε0^ω) = ψ0(Ωψ0(Ω+ψ0(Ω+1))) = ψ0(ψ1(ψ0(ψ1(0)))+ψ1(ψ0(ψ1(0)+ψ0(ψ1(0)+1))))

• [0{0 / 0{0 {0 / 1} 1 / 1}1}1] has level ε(ε0^ε0) = ψ0(Ωψ0(Ω+ψ0(Ω+ψ0(Ω)))) = ψ0(ψ1(ψ0(ψ1(0)))+ψ1(ψ0(ψ1(0)+ψ0(ψ1(0)+ψ0(ψ1(0))))))

• [0{0 / 0{0 / 2}1}1] has level ε(ε1) = ψ0(Ωψ0(Ω2)) = ψ0(ψ1(ψ0(ψ1(0)+ψ1(0))))

• [0{0 / 0{0 / 3}1}1] has level ε(ε2) = ψ0(Ωψ0(Ω3)) = ψ0(ψ1(ψ0(ψ1(0)+ψ1(0)+ψ1(0))))

• [0{0 / 0{0 / 0,1}1}1] has level ε(εω) = ψ0(Ωψ0(Ωω)) = ψ0(ψ1(ψ0(ψ1(1))))

• [0{0 / 0{0 / 0{1}1}1}1] has level ε(εω^ω) = ψ0(Ωψ0(Ωω^ω)) = ψ0(ψ1(ψ0(ψ1(ψ0(1)))))

• [0{0 / 0{0 / 0{0 / 1}1}1}1] has level ε(ε(ε0)) = ψ0(Ωψ0(Ωψ0(Ω))) = ψ0(ψ1(ψ0(ψ1(ψ0(ψ1(0))))))

• [0{0 / 0{0 / 0{0 / 0{0 / 1}1}1}1}1] has level ε(ε(ε(ε0))) = ψ0(Ωψ0(Ωψ0(Ωψ0(Ω)))) = ψ0(ψ1(ψ0(ψ1(ψ0(ψ1(ψ0(ψ1(0))))))))

• ...

Now we reach the new ordinal realm, ζ0 (Cantor's ordinal)!

• [0{0 / 0 / 1}1] has level ζ0 = ψ0(Ω^2) = ψ0(ψ1(ψ1(0)))

• [0{0 / 0 / 1}0,1] has level ζ0*ω = ω^(ζ0+1) = ψ0(Ω^2+1) = ψ0(ψ1(ψ1(0))+1)

• [0{0 / 0 / 1}0{0 / 0 / 1}1] has level ζ0^2 = ω^(ζ0*2) = ψ0(Ω^2+ψ0(Ω^2)) = ψ0(ψ1(ψ1(0))+ψ0(ψ1(ψ1(0))))

• [0{1 / 0 / 1}1] has level ζ0^ω = ω^ω^(ζ0+1) = ψ0(Ω^2+ψ0(Ω^2+1)) = ψ0(ψ1(ψ1(0))+ψ0(ψ1(ψ1(0))+1))

• [0{0 {0 / 0 / 1} 1 / 0 / 1}1] has level ζ0^ζ0 = ω^ω^(ζ0*2) = ψ0(Ω^2+ψ0(Ω^2+ψ0(Ω^2))) = ψ0(ψ1(ψ1(0))+ψ0(ψ1(ψ1(0))+ψ0(ψ1(ψ1(0)))))

• [0{0 {0 / 0 / 1} 0,1 / 0 / 1}1] has level ζ0^ζ0^ω = ω^ω^ω^(ζ0*2) = ψ0(Ω^2+ψ0(Ω^2+ψ0(Ω^2+1))) = ψ0(ψ1(ψ1(0))+ψ0(ψ1(ψ1(0))+ψ0(ψ1(ψ1(0))+1)))

• [0{0 / 1 / 1}1] has level ε(ζ0+1) = ψ0(Ω^2+Ω) = ψ0(ψ1(ψ1(0))+ψ1(0))

• [0{0 / 1 / 1}0,1] has level ω^(ε(ζ0+1)+1) = ψ0(Ω^2+Ω+1) = ψ0(ψ1(ψ1(0))+ψ1(0)+1)

• [0{0 / 2 / 1}1] has level ε(ζ0+2) = ψ0(Ω^2+Ω2) = ψ0(ψ1(ψ1(0))+ψ1(0)+ψ1(0))

• [0{0 / 0,1 / 1}1] has level ε(ζ0+ω) = ψ0(Ω^2+Ωω) = ψ0(ψ1(ψ1(0))+ψ1(1))

• [0{0 / 0 {0 / 1} 1 / 1}1] has level ε(ζ0+ε0) = ψ0(Ω^2+Ωψ0(Ω)) = ψ0(ψ1(ψ1(0))+ψ1(ψ0(ψ1(0))))

• [0{0 / 0 {0 / 0 / 1} 1 / 1}1] has level ε(ζ0*2) = ψ0(Ω^2+Ωψ0(Ω^2)) = ψ0(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(0)))))

• [0{0 / 0 {0 / 1 / 1} 1 / 1}1] has level ε(ε(ζ0+1)) = ψ0(Ω^2+Ωψ0(Ω^2+Ω)) = ψ0(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(0))+ψ1(0))))

• [0{0 / 0 {0 / 0,1 / 1} 1 / 1}1] has level ε(ε(ζ0+ω)) = ψ0(Ω^2+Ωψ0(Ω^2+Ωω)) = ψ0(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(0))+ψ1(1))))

• [0{0 / 0 {0 / 0 {0 / 0 / 1} 1 / 1} 1 / 1}1] has level ε(ε(ζ0*2)) = ψ0(Ω^2+Ωψ0(Ω^2+Ωψ0(Ω^2))) = ψ0(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(0)))))))

• [0{0 / 0 {0 / 0 {0 / 1 / 1} 1 / 1} 1 / 1}1] has level ε(ε(ε(ζ0+1))) = ψ0(Ω^2+Ωψ0(Ω^2+Ωψ0(Ω^2+Ω))) = ψ0(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(0))+ψ1(0))))))

• [0{0 / 0 / 2}1] has level ζ1 = ψ0(Ω^2*2) = ψ0(ψ1(ψ1(0))+ψ1(ψ1(0)))

• [0{0 / 1 / 2}1] has level ε(ζ1+1) = ψ0(Ω^2*2+Ω) = ψ0(ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(0))

• [0{0 / 0 / 3}1] has level ζ2 = ψ0(Ω^2*3) = ψ0(ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(ψ1(0)))

• [0{0 / 0 / 0,1}1] has level ζω = ψ0(Ω^2*ω) = ψ0(ψ1(ψ1(0)+1))

• [0{0 / 0 / 0{1}1}1] has level ζ(ω^ω) = ψ0(Ω^2*ω^ω) = ψ0(ψ1(ψ1(0)+ψ0(1)))

• [0{0 / 0 / 0{0 / 1}1}1] has level ζ(ε0) = ψ0(Ω^2*ψ0(Ω)) = ψ0(ψ1(ψ1(0)+ψ0(ψ1(0))))

• [0{0 / 0 / 0{0 / 0 / 1}1}1] has level ζ(ζ0) = ψ0(Ω^2*ψ0(Ω^2)) = ψ0(ψ1(ψ1(0)+ψ0(ψ1(ψ1(0)))))

• [0{0 / 0 / 0{0 / 0 / 2}1}1] has level ζ(ζ1) = ψ0(Ω^2*ψ0(Ω^2*2)) = ψ0(ψ1(ψ1(0)+ψ0(ψ1(ψ1(0))+ψ1(ψ1(0)))))

• [0{0 / 0 / 0{0 / 0 / 0,1}1}1] has level ζ(ζω) = ψ0(Ω^2*ψ0(Ω^2*ω)) = ψ0(ψ1(ψ1(0)+ψ0(ψ1(ψ1(0)+1))))

• [0{0 / 0 / 0{0 / 0 / 0{0 / 0 / 1}1}1}1] has level ζ(ζ(ζ0)) = ψ0(Ω^2*ψ0(Ω^2*ψ0(Ω^2))) = ψ0(ψ1(ψ1(0)+ψ0(ψ1(ψ1(0)+ψ0(ψ1(ψ1(0)))))))

• ...

• [0{0 / 0 / 0 / 1}1] has level η0 = φ(3,0) = ψ0(Ω^3) = ψ0(ψ1(ψ1(0)+ψ1(0)))

• [0{0 / 1 / 0 / 1}1] has level ε(η0+1) = ψ0(Ω^3+Ω) = ψ0(ψ1(ψ1(0)+ψ1(0))+ψ1(0))

• [0{0 / 0,1 / 0 / 1}1] has level ε(η0+ω) = ψ0(Ω^3+Ωω) = ψ0(ψ1(ψ1(0)+ψ1(0))+ψ1(1))

• [0{0 / 0{0 / 0 / 0 / 1}1 / 0 / 1}1] has level ε(η0*2) = ψ0(Ω^3+Ωψ0(Ω^3)) = ψ0(ψ1(ψ1(0)+ψ1(0))+ψ1(ψ0(ψ1(ψ1(0)+ψ1(0)))))

• [0{0 / 0 / 1 / 1}1] has level ζ(η0+1) = ψ0(Ω^3+Ω^2) = ψ0(ψ1(ψ1(0)+ψ1(0))+ψ1(ψ1(0)))

• [0{0 / 0 / 0,1 / 1}1] has level ζ(η0+ω) = ψ0(Ω^3+Ω^2*ω) = ψ0(ψ1(ψ1(0)+ψ1(0))+ψ1(ψ1(0)+1))

• [0{0 / 0 / 0{0 / 0 / 0 / 1}1 / 1}1] has level ζ(η0*2) = ψ0(Ω^3+Ω^2*ψ0(Ω^3)) = ψ0(ψ1(ψ1(0)+ψ1(0))+ψ1(ψ1(0)+ψ0(ψ1(ψ1(0)+ψ1(0)))))

• [0{0 / 0 / 0 / 2}1] has level η1 = ψ0(Ω^3*2) = ψ0(ψ1(ψ1(0)+ψ1(0))+ψ1(ψ1(0)+ψ1(0)))

• [0{0 / 0 / 0 / 0,1}1] has level ηω = ψ0(Ω^3*ω) = ψ0(ψ1(ψ1(0)+ψ1(0)+1))

• [0{0 / 0 / 0 / 0{0 / 0 / 0 / 1}1}1] has level η(η0) = ψ0(Ω^3*ψ0(Ω^3)) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ0(ψ1(ψ1(0)+ψ1(0)))))

• [0{0 / 0 / 0 / 0 / 1}1] has level φ(4,0) = ψ0(Ω^4) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0)))

• [0{0 / 1 / 0 / 0 / 1}1] has level ε(φ(4,0)+1) = ψ0(Ω^4+Ω) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0))+ψ1(0))

• [0{0 / 0 / 1 / 0 / 1}1] has level ζ(φ(4,0)+1) = ψ0(Ω^4+Ω^2) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0))+ψ1(ψ1(0)))

• [0{0 / 0 / 0 / 1 / 1}1] has level η(φ(4,0)+1) = ψ0(Ω^4+Ω^3) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0))+ψ1(ψ1(0)+ψ1(0)))

• [0{0 / 0 / 0 / 0 / 2}1] has level φ(4,1) = ψ0(Ω^4*2) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0))+ψ1(ψ1(0)+ψ1(0)+ψ1(0)))

• [0{0 / 0 / 0 / 0 / 0,1}1] has level φ(4,ω) = ψ0(Ω^4*ω) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0)+1))

• [0{0 / 0 / 0 / 0 / 0{0 / 0 / 0 / 0 / 1}1}1] has level φ(4,φ(4,0)) = ψ0(Ω^4*ψ0(Ω^4)) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0)+ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0))))

• [0{0 / 0 / 0 / 0 / 0 / 1}1] has level φ(5,0) = ψ0(Ω^5) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0)))

• [0{0 / 0 / 0 / 0 / 0 / 0,1}1] has level φ(5,ω) = ψ0(Ω^5*ω) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0)+1))

• [0{0 / 0 / 0 / 0 / 0 / 0 / 1}1] has level φ(6,0) = ψ0(Ω^6) = ψ0(ψ1(ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0)))

Now we introduce the separators of {}/!

• [0{0 {1}/ 1}1] has level φ(ω,0) = ψ0(Ω^ω) = ψ0(ψ1(ψ1(1)))

• [0{0 / 1 {1}/ 1}1] has level ε(φ(ω,0)+1) = ψ0(Ω^ω+Ω) = ψ0(ψ1(ψ1(1))+ψ1(0))

• [0{0 / 0 / 1 {1}/ 1}1] has level ζ(φ(ω,0)+1) = ψ0(Ω^ω+Ω^2) = ψ0(ψ1(ψ1(1))+ψ1(ψ1(0)))

• [0{0 {1}/ 2}1] has level φ(ω,1) = ψ0(Ω^ω*2) = ψ0(ψ1(ψ1(1))+ψ1(ψ1(1)))

• [0{0 {1}/ 3}1] has level φ(ω,2) = ψ0(Ω^ω*3) = ψ0(ψ1(ψ1(1))+ψ1(ψ1(1))+ψ1(ψ1(1)))

• [0{0 {1}/ 0,1}1] has level φ(ω,ω) = ψ0(Ω^ω*ω) = ψ0(ψ1(ψ1(1)+1))

• [0{0 {1}/ 0 {0 / 1} 1}1] has level φ(ω,ε0) = ψ0(Ω^ω*ψ0(Ω)) = ψ0(ψ1(ψ1(1)+ψ0(ψ1(0))))

• [0{0 {1}/ 0 {0 / 0 / 1} 1}1] has level φ(ω,ζ0) = ψ0(Ω^ω*ψ0(Ω^2)) = ψ0(ψ1(ψ1(1)+ψ0(ψ1(ψ1(0)))))

• [0{0 {1}/ 0 {0 {1}/ 1} 1}1] has level φ(ω,φ(ω,0)) = ψ0(Ω^ω*ψ0(Ω^ω)) = ψ0(ψ1(ψ1(1)+ψ0(ψ1(ψ1(1)))))

• [0{0 {1}/ 0 / 1}1] has level φ(ω+1,0) = ψ0(Ω^(ω+1)) = ψ0(ψ1(ψ1(1)+ψ1(0)))

• [0{0 {1}/ 1 / 1}1] has level φ(ω,φ(ω+1,0)+1) = ψ0(Ω^(ω+1)+Ω^ω) = ψ0(ψ1(ψ1(1)+ψ1(0))+ψ1(ψ1(1)))

• [0{0 {1}/ 0 / 2}1] has level φ(ω+1,1) = ψ0(Ω^(ω+1)*2) = ψ0(ψ1(ψ1(1)+ψ1(0))+ψ1(ψ1(1)+ψ1(0)))

• [0{0 {1}/ 0 / 0,1}1] has level φ(ω+1,ω) = ψ0(Ω^(ω+1)*2) = ψ0(ψ1(ψ1(1)+ψ1(0)+1))

• [0{0 {1}/ 0 / 0 / 1}1] has level φ(ω+2,0) = ψ0(Ω^(ω+2)) = ψ0(ψ1(ψ1(1)+ψ1(0)+ψ1(0)))

• [0{0 {1}/ 0 / 0 / 0 / 1}1] has level φ(ω+3,0) = ψ0(Ω^(ω+3)) = ψ0(ψ1(ψ1(1)+ψ1(0)+ψ1(0)+ψ1(0)))

• [0{0 {1}/ {1}/ 1}1] has level φ(ω2,0) = ψ0(Ω^(ω2)) = ψ0(ψ1(ψ1(1)+ψ1(1)))

• [0{0 {1}/ {1}/ 0 / 1}1] has level φ(ω2+1,0) = ψ0(Ω^(ω2+1)) = ψ0(ψ1(ψ1(1)+ψ1(1)+ψ1(0)))

• [0{0 {1}/ {1}/ 0 {1}/ 1}1] has level φ(ω3,0) = ψ0(Ω^(ω3)) = ψ0(ψ1(ψ1(1)+ψ1(1)+ψ1(1)))

• [0{0 {2}/ 1}1] has level φ(ω^2,0) = ψ0(Ω^(ω^2)) = ψ0(ψ1(ψ1(2)))

• [0{0 {3}/ 1}1] has level φ(ω^3,0) = ψ0(Ω^(ω^3)) = ψ0(ψ1(ψ1(3)))

• [0{0 {0,1}/ 1}1] has level φ(ω^ω,0) = ψ0(Ω^(ω^ω)) = ψ0(ψ1(ψ1(ψ0(1))))

• [0{0 {0,0,1}/ 1}1] has level φ(ω^ω^2,0) = ψ0(Ω^(ω^ω^2)) = ψ0(ψ1(ψ1(ψ0(2))))

• [0{0 {0{1}1}/ 1}1] has level φ(ω^ω^ω,0) = ψ0(Ω^(ω^ω^ω)) = ψ0(ψ1(ψ1(ψ0(ψ0(1)))))

• [0{0 {0{0,1}1}/ 1}1] has level φ(ω^ω^ω^ω,0) = ψ0(Ω^(ω^ω^ω^ω)) = ψ0(ψ1(ψ1(ψ0(ψ0(ψ0(1))))))

• ...

• [0{0 {0{0 / 1}1}/ 1}1] has level φ(ε0,0) = ψ0(Ω^ψ0(Ω)) = ψ0(ψ1(ψ1(ψ0(ψ1(0)))))

• [0{0 {0{0 / 2}1}/ 1}1] has level φ(ε1,0) = ψ0(Ω^ψ0(Ω2)) = ψ0(ψ1(ψ1(ψ0(ψ1(0)+ψ1(0)))))

• [0{0 {0{0 / 0,1}1}/ 1}1] has level φ(εω,0) = ψ0(Ω^ψ0(Ωω)) = ψ0(ψ1(ψ1(ψ0(ψ1(1)))))

• [0{0 {0{0 / 0 / 1}1}/ 1}1] has level φ(ζ0,0) = ψ0(Ω^ψ0(Ω^2)) = ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(0))))))

• [0{0 {0{0 / 0 / 0 / 1}1}/ 1}1] has level φ(η0,0) = ψ0(Ω^ψ0(Ω^3)) = ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(0)+ψ1(0))))))

• [0{0 {0{0 {1}/ 1}1}/ 1}1] has level φ(φ(ω,0),0) = ψ0(Ω^ψ0(Ω^ω)) = ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(1))))))

• [0{0 {0{0 {0,1}/ 1}1}/ 1}1] has level φ(φ(ω^ω,0),0) = ψ0(Ω^ψ0(Ω^(ω^ω))) = ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(ψ0(1)))))))

• [0{0 {0{0 {0{0 / 1}1}/ 1}1}/ 1}1] has level φ(φ(ε0,0),0) = ψ0(Ω^ψ0(Ω^ψ0(Ω))) = ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(ψ0(ψ1(0))))))))

• [0{0 {0{0 {0{0 / 0 / 1}1}/ 1}1}/ 1}1] has level φ(φ(ζ0,0),0) = ψ0(Ω^ψ0(Ω^ψ0(Ω^2))) = ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(0)))))))))

• [0{0 {0{0 {0{0 {1}/ 1}1}/ 1}1}/ 1}1] has level φ(φ(φ(ω,0),0),0) = ψ0(Ω^ψ0(Ω^ψ0(Ω^ω))) = ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(1)))))))))

• [0{0 {0{0 {0{0 {0 / 1}/ 1}1}/ 1}1}/ 1}1] has level φ(φ(φ(ε0,0),0),0) = ψ0(Ω^ψ0(Ω^ψ0(Ω^ψ0(Ω)))) = ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(ψ0(ψ1(ψ1(ψ0(ψ1(0)))))))))))

• ...

And finally, you reached the nested legion separator level, which has the FGH level of Γ0 (Feferman-Schütte ordinal)!

• [0{0 {0 / 1}/ 1}1] has level Γ0 = φ(1,0,0) = ψ0(Ω^Ω) = ψ0(ψ1(ψ1(ψ1(0))))

• [0{0 / 1 {0 / 1}/ 1}1] has level ε(Γ0+1) = φ(1,φ(1,0,0)+1) = ψ0(Ω^Ω+Ω) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(0))

• [0{0 / 0,1 {0 / 1}/ 1}1] has level ε(Γ0+ω) = φ(1,φ(1,0,0)+ω) = ψ0(Ω^Ω+Ωω) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(1))

• [0{0 / 0 {0 {0 / 1}/ 1} 1 {0 / 1}/ 1}1] has level ε(Γ0*2) = φ(1,φ(1,0,0)2) = ψ0(Ω^Ω+Ωψ0(Ω^Ω)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ0(ψ1(ψ1(ψ1(0))))))

• [0{0 / 0 / 1 {0 / 1}/ 1}1] has level ζ(Γ0+1) = φ(2,φ(1,0,0)+1) = ψ0(Ω^Ω+Ω^2) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(0)))

• [0{0 / 0 / 0 / 1 {0 / 1}/ 1}1] has level η(Γ0+1) = φ(3,φ(1,0,0)+1) = ψ0(Ω^Ω+Ω^3) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(0)+ψ1(0)))

• [0{0 {1}/ 1 {0 / 1}/ 1}1] has level φ(ω,Γ0+1) = φ(ω,φ(1,0,0)+1) = ψ0(Ω^Ω+Ω^ω) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(1)))

• [0{0 {0,1}/ 1 {0 / 1}/ 1}1] has level φ(ω^ω,Γ0+1) = φ(ω^ω,φ(1,0,0)+1) = ψ0(Ω^Ω+Ω^(ω^ω)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(1))))

• [0{0 {0 {0 / 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(ε0,Γ0+1) = φ(φ(1,0),φ(1,0,0)+1) = ψ0(Ω^Ω+Ω^ψ0(Ω)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(0)))))

• [0{0 {0 {0 / 0 / 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(ζ0,Γ0+1) = φ(φ(2,0),φ(1,0,0)+1) = ψ0(Ω^Ω+Ω^ψ0(Ω^2)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(0))))))

• [0{0 {0 {0 {1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(φ(ω,0),Γ0+1) = φ(φ(ω,0),φ(1,0,0)+1) = ψ0(Ω^Ω+Ω^ψ0(Ω^ω)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(1))))))

• [0{0 {0 {0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(Γ0,1) = φ(φ(1,0,0),1) = ψ0(Ω^Ω+Ω^ψ0(Ω^Ω)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))))))

• [0{0 {0 {0 {0 / 1}/ 1} 1}/ 2 {0 / 1}/ 1}1] has level φ(Γ0,2) = φ(φ(1,0,0),2) = ψ0(Ω^Ω+Ω^ψ0(Ω^Ω)2) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0))))))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))))))

• [0{0 {0 {0 {0 / 1}/ 1} 1}/ 0,1 {0 / 1}/ 1}1] has level φ(Γ0,ω) = φ(φ(1,0,0),ω) = ψ0(Ω^Ω+Ω^ψ0(Ω^Ω)ω) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))))+1))

• [0{0 {0 {0 {0 / 1}/ 1} 1}/ 0 {0 {0 {0 {0 / 1}/ 1} 1}/ 1} 1 {0 / 1}/ 1}1] has level φ(Γ0,Γ0) = φ(φ(1,0,0),φ(1,0,0)) = ψ0(Ω^Ω+Ω^ψ0(Ω^Ω)*ψ0(Ω^Ω)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))))+ψ0(ψ1(ψ1(ψ1(0))))))

• [0{0 {0 {0 {0 / 1}/ 1} 1}/ 0 / 1 {0 / 1}/ 1}1] has level φ(Γ0+1,0) = φ(φ(1,0,0)+1,0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω)+1)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0))))+1)))

• [0{0 {0 {0 {0 / 1}/ 1} 1}/ 0 / 0 / 1 {0 / 1}/ 1}1] has level φ(Γ0+2,0) = φ(φ(1,0,0)+2,0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω)+2)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0))))+2)))

• [0{0 {0 {0 {0 / 1}/ 1} 1}/ 0 {1}/ 1 {0 / 1}/ 1}1] has level φ(Γ0+ω,0) = φ(φ(1,0,0)+ω,0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω)+ω)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0))))+ψ0(1))))

• [0{0 {0 {0 {0 / 1}/ 1} 1}/ 0 {0 {0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(Γ0*2,0) = φ(φ(1,0,0)*2,0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω)*2)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0))))+ψ0(ψ1(ψ1(ψ1(0)))))))

• [0{0 {1 {0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(Γ0*ω,0) = φ(φ(1,0,0)*ω,0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω+1))) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))+1))))

• [0{0 {0 {0 {0 / 1}/ 1} 2}/ 1 {0 / 1}/ 1}1] has level φ(Γ0^2,0) = φ(φ(1,0,0)^2,0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω+ψ0(Ω^Ω)))) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))+ψ0(ψ1(ψ1(ψ1(0))))))))

• [0{0 {0 {0 {0 / 1}/ 1} 0,1}/ 1 {0 / 1}/ 1}1] has level φ(Γ0^ω,0) = φ(φ(1,0,0)^ω,0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω+ψ0(Ω^Ω+1)))) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))+ψ0(ψ1(ψ1(ψ1(0)))+1)))))

• [0{0 {0 {0 {0 / 1}/ 1} 0 {0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(Γ0^Γ0,0) = φ(φ(1,0,0)^φ(1,0,0),0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω+ψ0(Ω^Ω+ψ0(Ω^Ω))))) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))+ψ0(ψ1(ψ1(ψ1(0)))+ψ0(ψ1(ψ1(ψ1(0)))))))))

• [0{0 {0 {1 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(Γ0^Γ0^ω,0) = φ(φ(1,0,0)^φ(1,0,0)^ω,0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω+ψ0(Ω^Ω+ψ0(Ω^Ω+1))))) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))+ψ0(ψ1(ψ1(ψ1(0)))+ψ0(ψ1(ψ1(ψ1(0)))+1))))))

• [0{0 {0 {0 / 1 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(ε(Γ0+1),0) = φ(φ(1,φ(1,0,0)+1),0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω+Ω))) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))+ψ1(0)))))

• [0{0 {0 {0 {0 {0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(φ(Γ0,1),0) = φ(φ(φ(1,0,0),1),0) = ψ0(Ω^Ω+Ω^(ψ0(Ω^Ω+Ω^ψ0(Ω^Ω)))) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0))))))))))

• ...

Oh my complex separators! Moving on:

• [0{0 {0 / 1}/ 2}1] has level Γ1 = φ(1,0,1) = ψ0(Ω^Ω*2) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))))

• [0{0 {0 {0 {0 / 1}/ 2} 1}/ 1 {0 / 1}/ 2}1] has level φ(Γ1,1) = φ(φ(1,0,1),1) = ψ0(Ω^Ω*2+Ω^ψ0(Ω^Ω*2)) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0)))))))

• [0{0 {0 / 1}/ 3}1] has level Γ2 = φ(1,0,2) = ψ0(Ω^Ω*3) = ψ0(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))))

• [0{0 {0 / 1}/ 0,1}1] has level Γω = φ(1,0,ω) = ψ0(Ω^Ω*ω) = ψ0(ψ1(ψ1(ψ1(0))+1))

• [0{0 {0 / 1}/ 0 {0 / 1}1} 1] has level Γ(ε0) = φ(1,0,φ(1,0)) = ψ0(Ω^Ω*ψ0(Ω)) = ψ0(ψ1(ψ1(ψ1(0))+ψ0(ψ1(0))))

• [0{0 {0 / 1}/ 0 {0 {0 / 1}/ 1} 1}1] has level Γ(Γ0) = φ(1,0,φ(1,0,0)) = ψ0(Ω^Ω*ψ0(Ω^Ω)) = ψ0(ψ1(ψ1(ψ1(0))+ψ0(ψ1(ψ1(ψ1(0))))))

• [0{0 {0 / 1}/ 0 {0 {0 / 1}/ 0 {0 {0 / 1}/ 1} 1} 1}1] has level Γ(Γ(Γ0)) = φ(1,0,φ(1,0,φ(1,0,0))) = ψ0(Ω^Ω*ψ0(Ω^Ω*ψ0(Ω^Ω))) = ψ0(ψ1(ψ1(ψ1(0))+ψ0(ψ1(ψ1(ψ1(0))+ψ0(ψ1(ψ1(ψ1(0))))))))

• ...

• [0{0 {0 / 1}/ 0 / 1}1] has level φ(1,1,0) = ψ0(Ω^(Ω+1)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(0)))

• [0{0 {0 / 1}/ 1 / 1}1] has level Γ(φ(1,1,0)+1) = ψ0(Ω^(Ω+1)+Ω^Ω) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(0))+ψ1(ψ1(ψ1(0))))

• [0{0 {0 / 1}/ 0 / 2}1] has level φ(1,1,1) = ψ0(Ω^(Ω+1)*2) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(0))+ψ1(ψ1(ψ1(0))+ψ1(0)))

• [0{0 {0 / 1}/ 0 / 3}1] has level φ(1,1,2) = ψ0(Ω^(Ω+1)*3) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(0))+ψ1(ψ1(ψ1(0))+ψ1(0))+ψ1(ψ1(ψ1(0))+ψ1(0)))

• [0{0 {0 / 1}/ 0 / 0,1}1] has level φ(1,1,ω) = ψ0(Ω^(Ω+1)*ω) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(0)+1))

• [0{0 {0 / 1}/ 0 / 0 {0 {0 / 1}/ 0 / 1} 1}1] has level φ(1,1,φ(1,1,0)) = ψ0(Ω^(Ω+1)*ψ0(Ω^(Ω+1))) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(0)+ψ0(ψ1(ψ1(ψ1(0))+ψ1(0)))))

• [0{0 {0 / 1}/ 0 / 0 / 1}1] has level φ(1,2,0) = ψ0(Ω^(Ω+2)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(0)+ψ1(0)))

• [0{0 {0 / 1}/ 0 / 0 / 0 / 1}1] has level φ(1,3,0) = ψ0(Ω^(Ω+3)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(0)+ψ1(0)+ψ1(0)))

• [0{0 {0 / 1}/ 0 {1}/ 1}1] has level φ(1,ω,0) = ψ0(Ω^(Ω+ω)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(1)))

• [0{0 {0 / 1}/ 0 {0,1}/ 1}1] has level φ(1,ω^ω,0) = ψ0(Ω^(Ω+ω^ω)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ0(1))))

• [0{0 {0 / 1}/ 0 {0 {0 / 1} 1}/ 1}1] has level φ(1,ε0,0) = ψ0(Ω^(Ω+ψ0(Ω))) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ0(ψ1(0)))))

• [0{0 {0 / 1}/ 0 {0 {0 {0 / 1}/ 1} 1}/ 1}1] has level φ(1,Γ0,0) = ψ0(Ω^(Ω+ψ0(Ω^Ω))) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(ψ1(0)))))))

• [0{0 {0 / 1}/ 0 {0 {0 {0 / 1}/ 0 {0 {0 {0 / 1}/ 1} 1}/ 1} 1}/ 1}1] has level φ(1,φ(1,Γ0,0),0) = ψ0(Ω^(Ω+ψ0(Ω^(Ω+ψ0(Ω^(Ω+1)))))) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(ψ1(0))+ψ0(ψ1(ψ1(ψ1(0)))))))))

• ...

• [0{0 {0 / 1}/ 0 {0 / 1}/ 1}1] has level φ(2,0,0) = ψ0(Ω^(Ω2)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))))

• [0{0 {0 / 1}/ 1 {0 / 1}/ 1}1] has level Γ(φ(2,0,0)+1) = ψ0(Ω^(Ω2)+Ω^Ω) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))))

• [0{0 {0 / 1}/ 0 / 1 {0 / 1}/ 1}1] has level φ(1,1,φ(2,0,0)+1) = ψ0(Ω^(Ω2)+Ω^(Ω+1)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))+ψ1(0)))

• [0{0 {0 / 1}/ 0 / 0 / 1 {0 / 1}/ 1}1] has level φ(1,2,φ(2,0,0)+1) = ψ0(Ω^(Ω2)+Ω^(Ω+2)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))+ψ1(0)+ψ1(0)))

• [0{0 {0 / 1}/ 0 {1}/ 1 {0 / 1}/ 1}1] has level φ(1,ω,φ(2,0,0)+1) = ψ0(Ω^(Ω2)+Ω^(Ω+ω)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))+ψ1(1)))

• [0{0 {0 / 1}/ 0 {0 {0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(1,Γ0,φ(2,0,0)+1) = ψ0(Ω^(Ω2)+Ω^(Ω+ψ0(Ω^Ω))) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(ψ1(0)))))))

• [0{0 {0 / 1}/ 0 {0 {0 {0 / 1}/ 0 {0 / 1}/ 1} 1}/ 1 {0 / 1}/ 1}1] has level φ(1,φ(2,0,0),1) = ψ0(Ω^(Ω2)+Ω^(Ω+ψ0(Ω^(Ω2)))) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))))))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 2}1] has level φ(2,0,1) = ψ0(Ω^(Ω2)*2) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 0,1}1] has level φ(2,0,ω) = ψ0(Ω^(Ω2)*ω) = ψ0(ψ1(ψ1(ψ1(0))+1))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 0 {0 {0 / 1}/ 0 {0 / 1}/ 1} 1}1] has level φ(2,0,φ(2,0,0)) = ψ0(Ω^(Ω2)*ψ0(Ω^(Ω2))) = ψ0(ψ1(ψ1(ψ1(0))+ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))))))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 0 / 1}1] has level φ(2,1,0) = ψ0(Ω^(Ω2+1)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(0)))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 0 / 0 / 1}1] has level φ(2,2,0) = ψ0(Ω^(Ω2+2)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(0)+ψ1(0)))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 0 {1}/ 1}1] has level φ(2,ω,0) = ψ0(Ω^(Ω2+ω)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(1)))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 0 {0 {0 {0 / 1}/ 0 {0 / 1}/ 1} 1}/ 1}1] has level φ(2,φ(2,0,0),0) = ψ0(Ω^(Ω2+ψ0(Ω^(Ω2)))) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))))))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 0 {0 / 1}/ 1}1] has level φ(3,0,0) = ψ0(Ω^(Ω3)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(ψ1(0))))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 0 {0 / 1}/ 0 {0 / 1}/ 1}1] has level φ(4,0,0) = ψ0(Ω^(Ω4)) = ψ0(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(ψ1(0))+ψ1(ψ1(0))))

• ...

• [0{0 {1 / 1}/ 1}1] has level φ(ω,0,0) = ψ0(Ω^(Ωω)) = ψ0(ψ1(ψ1(ψ1(0)+1))))

• [0{0 / 1 {1 / 1}/ 1}1] has level ε(φ(ω,0,0)+1) = ψ0(Ω^(Ωω)+Ω) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(0)))

• [0{0 {0 / 1}/ 1 {1 / 1}/ 1}1] has level Γ(φ(ω,0,0)+1) = ψ0(Ω^(Ωω)+Ω^Ω) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(ψ1(ψ1(0)))))

• [0{0 {0 / 1}/ 0 {0 / 1} 1 {1 / 1}/ 1}1] has level φ(2,0,φ(ω,0,0)+1) = ψ0(Ω^(Ωω)+Ω^(Ω2)) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))))

• [0{0 {1 / 1}/ 2}1] has level φ(ω,0,1) = ψ0(Ω^(Ωω)*2) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(ψ1(ψ1(0)+1))))

• [0{0 {1 / 1}/ 0,1}1] has level φ(ω,0,ω) = ψ0(Ω^(Ωω)*ω) = ψ0(ψ1(ψ1(ψ1(0)+1))+1))

• [0{0 {1 / 1}/ 0 {0 {1 / 1}/ 1} 1}1] has level φ(ω,0,φ(ω,0,0)) = ψ0(Ω^(Ωω)*ψ0(Ω^(Ωω))) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ0(ψ1(ψ1(ψ1(0)+1))))))

• [0{0 {1 / 1}/ 0 / 1}1] has level φ(ω,1,0) = ψ0(Ω^(Ωω+1)) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(0)))

• [0{0 {1 / 1}/ 0 / 0 / 1}1] has level φ(ω,2,0) = ψ0(Ω^(Ωω+2)) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(0)+ψ1(0)))

• [0{0 {1 / 1}/ 0 {1}/ 1}1] has level φ(ω,ω,0) = ψ0(Ω^(Ωω+ω)) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(1)))

• [0{0 {1 / 1}/ 0 {0 {0 {1 / 1}/ 1} 1}/ 1}1] has level φ(ω,φ(ω,0,0),0) = ψ0(Ω^(Ωω+ψ0(Ω^(Ωω)))) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(ψ0(ψ1(ψ1(ψ1(0)+1)))))))

• [0{0 {1 / 1}/ 0 {0 / 1}/ 1}1] has level φ(ω+1,0,0) = ψ0(Ω^(Ωω+Ω)) = ψ0(ψ1(ψ1(ψ1(0)+1)+ψ1(ψ1(0)))))

• [0{0 {1 / 1}/ 0 {0 / 1}/ 0 {0 / 1}/ 1}1] has level φ(ω+2,0,0) = ψ0(Ω^(Ωω+Ω2)) = ψ0(ψ1(ψ1(ψ1(0)+1)+ψ1(ψ1(0))+ψ1(ψ1(0)))))

• [0{0 {1 / 1}/ 0 {1 / 1}/ 1}1] has level φ(ω2,0,0) = ψ0(Ω^(Ωω2)) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(ψ1(0)+1))))

• [0{0 {1 / 1}/ 0 {1 / 1}/ 0 {1 / 1}/ 1}1] has level φ(ω3,0,0) = ψ0(Ω^(Ωω3)) = ψ0(ψ1(ψ1(ψ1(0)+1))+ψ1(ψ1(0)+1))+ψ1(ψ1(0)+1))))

• [0{0 {2 / 1}/ 1}1] has level φ(ω^2,0,0) = ψ0(Ω^(Ωω^2)) = ψ0(ψ1(ψ1(ψ1(0)+2))))

• [0{0 {3 / 1}/ 1}1] has level φ(ω^3,0,0) = ψ0(Ω^(Ωω^3)) = ψ0(ψ1(ψ1(ψ1(0)+3))))

• [0{0 {0,1 / 1}/ 1}1] has level φ(ω^ω,0,0) = ψ0(Ω^(Ωω^ω)) = ψ0(ψ1(ψ1(ψ1(0)+ψ0(1)))))

• [0{0 {0 {0 / 1} 1 / 1}/ 1}1] has level φ(ε0,0,0) = ψ0(Ω^(Ωψ0(Ω))) = ψ0(ψ1(ψ1(ψ1(0)+ψ0(ψ1(0))))))

• [0{0 {0 {0 {0 / 1}/ 1} 1 / 1}/ 1}1] has level φ(Γ0,0,0) = ψ0(Ω^(Ωψ0(Ω^Ω))) = ψ0(ψ1(ψ1(ψ1(0)+ψ0(ψ1(ψ1(ψ1(0))))))))

• [0{0 {0 {0 {0 / 1}/ 1} 1 / 1}/ 1}1] has level φ(φ(Γ0,0,0),0,0) = ψ0(Ω^(Ωψ0(Ω^Ωψ0(Ω^Ω)))) = ψ0(ψ1(ψ1(ψ1(0)+ψ0(ψ1(ψ1(ψ1(0)+ψ0(ψ1(ψ1(ψ1(0)))))))))))

• ...

• [0{0 {0 / 2}/ 1}1] has level φ(1,0,0,0) = ψ0(Ω^Ω^2) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))) (Ackermann's ordinal)

• [0{0 / 1 {0 / 2}/ 1}1] has level ε(φ(1,0,0,0)+1) = ψ0(Ω^Ω^2+Ω) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(0))

• [0{0 / 0 / 1 {0 / 2}/ 1}1] has level ζ(φ(1,0,0,0)+1) = ψ0(Ω^Ω^2+Ω^2) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(ψ1(0)))

• [0{0 {1}/ 1 {0 / 2}/ 1}1] has level φ(ω,φ(1,0,0,0)+1) = ψ0(Ω^Ω^2+Ω^ω) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(ψ1(1)))

• [0{0 {0 {0 {0 / 2}/ 1} 1}/ 1 {0 / 2}/ 1}1] has level φ(φ(1,0,0,0),1) = ψ0(Ω^Ω^2+Ω^ψ0(Ω^Ω^2)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))))))

• [0{0 {0 / 1}/ 1 {0 / 2}/ 1}1] has level Γ(φ(1,0,0,0)+1) = ψ0(Ω^Ω^2+Ω^Ω) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(ψ1(ψ1(0))))

• [0{0 {0 / 1}/ 0 {0 / 1}/ 1 {0 / 2}/ 1}1] has level φ(2,0,φ(1,0,0,0)+1) = ψ0(Ω^Ω^2+Ω^(Ω2)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))))

• [0{0 {1 / 1}/ 1 {0 / 2}/ 1}1] has level φ(ω,0,φ(1,0,0,0)+1) = ψ0(Ω^Ω^2+Ω^(Ωω)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(ψ1(ψ1(0)+1)))

• [0{0 {0 {0 {0 / 2}/ 1} 1 / 1}/ 1 {0 / 2}/ 1}1] has level φ(φ(1,0,0,0),0,1) = ψ0(Ω^Ω^2+Ω^(Ωψ0(Ω^Ω^2))) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(ψ1(ψ1(0)+ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))))))

• [0 {0 / 2}/ 2}1] has level φ(1,0,0,1) = ψ0(Ω^Ω^2*2) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))+ψ1(ψ1(ψ1(0)+ψ1(0)))))

• [0 {0 / 2}/ 0,1}1] has level φ(1,0,0,ω) = ψ0(Ω^Ω^2*ω) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+1))

• [0 {0 / 2}/ 0 {0 {0 / 2}/ 1} 1}1] has level φ(1,0,0,φ(1,0,0,0)) = ψ0(Ω^Ω^2*ψ0(Ω^Ω^2))= ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))))))

• [0 {0 / 2}/ 0 / 1}1] has level φ(1,0,1,0) = ψ0(Ω^(Ω^2+1)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+ψ1(0)))

• [0 {0 / 2}/ 0 {1}/ 1}1] has level φ(1,0,ω,0) = ψ0(Ω^(Ω^2+ω)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+ψ1(1)))

• [0 {0 / 2}/ 0 {0 {0 {0 / 2}/ 1} 1}/ 1}1] has level φ(1,0,φ(1,0,0,0),0) = ψ0(Ω^(Ω^2+ψ0(Ω^Ω^2)))= ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+ψ1(ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))))))

• [0 {0 / 2}/ 0 {0 / 1}/ 1}1] has level φ(1,1,0,0) = ψ0(Ω^(Ω^2+Ω)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+ψ1(ψ1(0))))

• [0 {0 / 2}/ 0 {0 / 1}/ 0 {0 / 1}/ 1}1] has level φ(1,2,0,0) = ψ0(Ω^(Ω^2+Ω2)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+ψ1(ψ1(0))+ψ1(ψ1(0))))

• [0 {0 / 2}/ 0 {1 / 1}/ 1}1] has level φ(1,ω,0,0) = ψ0(Ω^(Ω^2+Ωω)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+ψ1(ψ1(0)+1)))

• [0 {0 / 2}/ 0 {1 / 1}/ 1}1] has level φ(1,φ(1,0,0,0),0,0) = ψ0(Ω^(Ω^2+Ωψ0(Ω^Ω^2))) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))+ψ1(ψ1(0)+ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))))))

• [0 {0 / 2}/ 0 {0 / 2}/ 1] has level φ(2,0,0,0) = ψ0(Ω^(Ω^2*2)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))+ψ1(ψ1(0)+ψ1(0)))))

• [0 {0 / 2}/ 0 {0 / 2}/ 0 {0 / 2}/ 1] has level φ(3,0,0,0) = ψ0(Ω^(Ω^2*3)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))+ψ1(ψ1(0)+ψ1(0))+ψ1(ψ1(0)+ψ1(0)))))

• [0 {1 / 2}/ 1] has level φ(ω,0,0,0) = ψ0(Ω^(Ω^2*ω)) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)+1))))

• [0 {0 {0 {0 / 2}/ 1} 1 / 2}/ 1] has level φ(φ(1,0,0,0),0,0,0) = ψ0(Ω^(Ω^2*ψ0(Ω^Ω^2))) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)+ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)))))))))

• [0 {0 / 3}/ 1] has level φ(1,0,0,0,0) = ψ0(Ω^Ω^3) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)+ψ1(0))))

• [0 {0 / 4}/ 1] has level φ(1,0,0,0,0,0) = ψ0(Ω^Ω^4) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0))))

• [0 {0 / 5}/ 1] has level φ(1,0,0,0,0,0,0) = ψ0(Ω^Ω^5) = ψ0(ψ1(ψ1(ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0)+ψ1(0))))

• ...

• [0 {0 / 0,1}/ 1] has level ψ0(Ω^Ω^ω) = ψ0(ψ1(ψ1(ψ1(1)))) (small Veblen ordinal)

• [0 / 1 {0 / 0,1}/ 1] has level ψ0(Ω^Ω^ω+Ω) = ψ0(ψ1(ψ1(ψ1(1)))+ψ1(0))

• [0 / 0 / 1 {0 / 0,1}/ 1] has level ψ0(Ω^Ω^ω+Ω^2) = ψ0(ψ1(ψ1(ψ1(1)))+ψ1(ψ1(0)))

• [0 {0 / 1}/ 1 {0 / 0,1}/ 1] has level ψ0(Ω^Ω^ω+Ω^Ω) = ψ0(ψ1(ψ1(ψ1(1)))+ψ1(ψ1(ψ1(0))))

• [0 {0 / 2}/ 1 {0 / 0,1}/ 1] has level ψ0(Ω^Ω^ω+Ω^Ω^2) = ψ0(ψ1(ψ1(ψ1(1)))+ψ1(ψ1(ψ1(0)+ψ1(0))))

• [0 {0 / 0,1}/ 2] has level ψ0(Ω^Ω^ω*2) = ψ0(ψ1(ψ1(ψ1(1)))+ψ1(ψ1(ψ1(1))))

• [0 {0 / 0,1}/ 0,1] has level ψ0(Ω^Ω^ω*ω) = ψ0(ψ1(ψ1(ψ1(1))+1))

• [0 {0 / 0,1}/ 0 / 1] has level ψ0(Ω^(Ω^ω+1)) = ψ0(ψ1(ψ1(ψ1(1))+ψ1(0)))

• [0 {0 / 0,1}/ 0 {0 / 0,1}/ 1] has level ψ0(Ω^(Ω^ω*2)) = ψ0(ψ1(ψ1(ψ1(1))+ψ1(ψ1(1))))

• [0 {1 / 0,1}/ 1] has level ψ0(Ω^(Ω^ω*ω)) = ψ0(ψ1(ψ1(ψ1(1)+1)))

• [0 {0 {0 {0 / 0,1}/ 1} 1 / 0,1}/ 1] has level ψ0(Ω^(Ω^ω*ψ0(Ω^Ω^ω))) = ψ0(ψ1(ψ1(ψ1(1)+ψ0(ψ1(ψ1(ψ1(1)))))))

• [0 {0 / 1,1}/ 1] has level ψ0(Ω^Ω^(ω+1)) = ψ0(ψ1(ψ1(ψ1(1)+ψ1(0))))

• [0 {0 / 2,1}/ 1] has level ψ0(Ω^Ω^(ω+2)) = ψ0(ψ1(ψ1(ψ1(1)+ψ1(0)+ψ1(0))))

• [0 {0 / 0,2}/ 1] has level ψ0(Ω^Ω^(ω2)) = ψ0(ψ1(ψ1(ψ1(1)+ψ1(1))))

• [0 {0 / 0,0,1}/ 1] has level ψ0(Ω^Ω^(ω^2)) = ψ0(ψ1(ψ1(ψ1(2))))

• [0 {0 / 0{1}1}/ 1] has level ψ0(Ω^Ω^(ω^ω)) = ψ0(ψ1(ψ1(ψ1(ψ0(1)))))

• [0 {0 / 0{0,1}1}/ 1] has level ψ0(Ω^Ω^(ω^ω^ω)) = ψ0(ψ1(ψ1(ψ1(ψ0(ψ0(1))))))

• [0 {0 / 0 {0 / 1} 1}/ 1] has level ψ0(Ω^Ω^ψ0(Ω)) = ψ0(ψ1(ψ1(ψ1(ψ0(ψ1(0))))))

• [0 {0 / 0 {0 / 0 / 1} 1}/ 1] has level ψ0(Ω^Ω^ψ0(Ω^2)) = ψ0(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(0)))))))

• [0 {0 / 0 {0 {1}/ 1} 1}/ 1] has level ψ0(Ω^Ω^ψ0(Ω^ω)) = ψ0(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(1)))))))

• [0 {0 / 0 {0 {0 / 1}/ 1} 1}/ 1] has level ψ0(Ω^Ω^ψ0(Ω^Ω)) = ψ0(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0))))))))

• [0 {0 / 0 {0 {0 / 2}/ 1} 1}/ 1] has level ψ0(Ω^Ω^ψ0(Ω^Ω^2)) = ψ0(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)+ψ1(0))))))))

• [0 {0 / 0 {0 {0 / 0,1}/ 1} 1}/ 1] has level ψ0(Ω^Ω^ψ0(Ω^Ω^ω)) = ψ0(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(1))))))))

• [0 {0 / 0 {0 {0 / 0 {0 {0 {0 / 1}/ 1} 1} 1}/ 1} 1}/ 1] has level ψ0(Ω^Ω^ψ0(Ω^Ω^ψ0(Ω^Ω))) = ψ0(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0))))))))))))

• ...

Now you reached the large Veblen ordinal level!

• [0 {0 {0 / 0 / 1}/ 1} 1] has level ψ0(Ω^Ω^Ω) = ψ0(ψ1(ψ1(ψ1(ψ1(0))))) (large Veblen ordinal)

• [0 {0 / 1 {0 / 0 / 1}/ 1} 1] has level ψ0(Ω^Ω^Ω+Ω) = ψ0(ψ1(ψ1(ψ1(ψ1(0))))+ψ1(0))

• [0 {0 / 0 / 1 {0 / 0 / 1}/ 1} 1] has level ψ0(Ω^Ω^Ω+Ω^2) = ψ0(ψ1(ψ1(ψ1(ψ1(0))))+ψ1(ψ1(0)))

• [0 {0 {0 / 1}/ 1 {0 / 0 / 1}/ 1} 1] has level ψ0(Ω^Ω^Ω+Ω^Ω) = ψ0(ψ1(ψ1(ψ1(ψ1(0))))+ψ1(ψ1(ψ1(0))))

• [0 {0 {0 / 0 / 1}/ 2} 1] has level ψ0(Ω^Ω^Ω*2) = ψ0(ψ1(ψ1(ψ1(ψ1(0))))+ψ1(ψ1(ψ1(ψ1(0)))))

• [0 {0 {0 / 0 / 1}/ 0,1} 1] has level ψ0(Ω^Ω^Ω*ω) = ψ0(ψ1(ψ1(ψ1(ψ1(0)))+1))

• [0 {0 {0 / 0 / 1}/ 0 / 1} 1] has level ψ0(Ω^(Ω^Ω+1)) = ψ0(ψ1(ψ1(ψ1(ψ1(0)))+ψ1(0)))

• [0 {0 {0 / 0 / 1}/ 0 {1}/ 1} 1] has level ψ0(Ω^(Ω^Ω+ω)) = ψ0(ψ1(ψ1(ψ1(ψ1(0)))+ψ1(1)))

• [0 {0 {0 / 0 / 1}/ 0 {0 / 1}/ 1} 1] has level ψ0(Ω^(Ω^Ω+Ω)) = ψ0(ψ1(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(0))))

• [0 {0 {0 / 0 / 1}/ 0 {0 / 0,1}/ 1} 1] has level ψ0(Ω^(Ω^Ω+Ω^ω)) = ψ0(ψ1(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(1))))

• [0 {0 {0 / 0 / 1}/ 0 {0 / 0 / 1}/ 1} 1] has level ψ0(Ω^(Ω^Ω*2)) = ψ0(ψ1(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0)))))

• [0 {0 {1 / 0 / 1}/ 1} 1] has level ψ0(Ω^(Ω^Ω*ω)) = ψ0(ψ1(ψ1(ψ1(ψ1(0))+1)))

• [0 {0 {0 / 1 / 1}/ 1} 1] has level ψ0(Ω^Ω^(Ω+1)) = ψ0(ψ1(ψ1(ψ1(ψ1(0))+ψ1(0))))

• [0 {0 {0 / 0,1 / 1}/ 1} 1] has level ψ0(Ω^Ω^(Ω+ω)) = ψ0(ψ1(ψ1(ψ1(ψ1(0))+ψ1(1))))

• [0 {0 {0 / 0 {0 {0 / 0 / 1}/ 1} 1 / 1}/ 1} 1] has level ψ0(Ω^Ω^(Ω+ψ0(Ω^Ω^Ω))) = ψ0(ψ1(ψ1(ψ1(ψ1(0))+ψ1(ψ0(ψ1(ψ1(ψ1(ψ1(0)))))))))

• [0 {0 {0 / 0 / 2}/ 1} 1] has level ψ0(Ω^Ω^(Ω2)) = ψ0(ψ1(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0)))))

• [0 {0 {0 / 0 / 0,1}/ 1} 1] has level ψ0(Ω^Ω^(Ωω)) = ψ0(ψ1(ψ1(ψ1(ψ1(0)+1))))

• [0 {0 {0 / 0 / 0 {0 {0 / 0 / 1}/ 1} 1}/ 1} 1] has level ψ0(Ω^Ω^(Ωψ0(Ω^Ω^Ω))) = ψ0(ψ1(ψ1(ψ1(ψ1(0)+ψ0(ψ1(ψ1(ψ1(ψ1(0)))))))))

• [0 {0 {0 / 0 / 0 / 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^2) = ψ0(ψ1(ψ1(ψ1(ψ1(0)+ψ1(0)))))

• [0 {0 {0 / 0 / 0 / 0 / 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^3) = ψ0(ψ1(ψ1(ψ1(ψ1(0)+ψ1(0)+ψ1(0)))))

• [0 {0 {0 {1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^ω) = ψ0(ψ1(ψ1(ψ1(ψ1(1)))))

• [0 {0 {0 {0 {0 {0 / 1}/ 1} 1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^ψ0(Ω^Ω)) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(0)))))))))

• [0 {0 {0 {0 {0 {0 / 0 / 1}/ 1} 1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^ψ0(Ω^Ω^Ω)) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ0(ψ1(ψ1(ψ1(ψ1(0))))))))))

• [0 {0 {0 {0 / 1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^Ω) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(0))))))

• [0 {0 {0 {0 / 1}/ 2}/ 1} 1] has level ψ0(Ω^Ω^(Ω^Ω*2)) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(0)))+ψ1(ψ1(ψ1(0))))))

• [0 {0 {0 {0 / 1}/ 0,1}/ 1} 1] has level ψ0(Ω^Ω^(Ω^Ω*ω)) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(0))+1))))

• [0 {0 {0 {0 / 1}/ 0 / 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^(Ω+1)) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(0))+ψ1(0)))))

• [0 {0 {0 {0 / 1}/ 0 {0 / 1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^(Ω2)) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(0))+ψ1(ψ1(0))))))

• [0 {0 {0 {1 / 1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^(Ωω)) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(0)+1)))))

• [0 {0 {0 {0 / 2}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^Ω^2) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(0)+ψ1(0))))))

• [0 {0 {0 {0 / 0,1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^Ω^ω) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(1))))))

• [0 {0 {0 {0 / 0 / 1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^Ω^Ω) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(ψ1(0)))))))

• [0 {0 {0 {0 {0 / 1}/ 1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^Ω^Ω^Ω) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(ψ1(ψ1(0))))))))

• [0 {0 {0 {0 {0 / 0 / 1}/ 1}/ 1}/ 1} 1] has level ψ0(Ω^Ω^Ω^Ω^Ω^Ω^Ω) = ψ0(ψ1(ψ1(ψ1(ψ1(ψ1(ψ1(ψ1(ψ1(0)))))))))

As you can see that each additional layer of the / nestings adds two Ω's into the ordinal power in the Buchholz's function (systemically, using system of fundamental sequences based on (extended) Buchholz's function, this adds two ψ1 nestings in the Buchholz's normal form), and hence the limit is ψ0(ε(Ω+1)) = ψ0(Ω_2) = ψ0(ψ2(0)) with respect to the Buchholz's function (the Bachmann-Howard ordinal).