My last notation could beyond ψ(I),but it's too confusion,both growth level and definition.

I tried to get a better definition which could be more clear.

To get more attention,I want to introduce a weak but easy to understand subset of my notation v2 -- without using the symbol comma.

It's much more clear and easier to understand,so it's too weak that it can just reach the level Γ_0.

(Sometimes we will regard BS or LBS as an adjective.)

1.We define a set BS (means bracket-string):

(1)Empty string φ is in BS.

(2)If X(1),X(2) is in BS,then X(1)X(2) and [X(1)] (so [X(2)]) is in BS.

Now every BS X can be written in the form [X(1)][X(2)][X(3)]...[X(n)].

For BS X,we define n(X)=n,and X(i) as we have written.

For a closed interval [a,b],0≤a,b≤n,define X([a,b])=[X(a)][X(a+1)]...[X(b)].If b<a,X([a,b])=φ.

2.We define a subset LBS (means legal bracket-string),a map >>: N*LBS -> LBS and a map S: LBS -> Power(BS).

These three objects must satisfied that:

(1)φ is LBS.n>>φ=φ for any n belongs to N.

(2)If X is LBS,then X[] is in LBS,n>>X[]=X for any n belongs to N.

(3)If X is LBS,S(0) is the set {X},S(i) is the set N>>S(i-1) for positive integer i,then S(X)=union of S(i),i=0,1,2,3,...

(4)For any BS X,satisfy these below three situation:

((1))X≠φ,n=n(X),every X(i) is LBS.

((2))The biggest i s.t. X(n) doesn't belong to S(X(i)) exists and satisfy 0>>X(n) belongs to S(X(i)).

((3))Define m to be the biggest number s.t. m>>X(n) belongs to S(X(i)).If m is infinity,set m=1 instead of x.(In fact,you can choose any positive integer.)

Define Y(0)=X([1,i-1]),Y(1)=X([1,n-1]),Y(i+1)=Y(i)[m+i>>X(n)]X([i+1,n-1]) for any positive integer i.

For any i belongs to N we have Y(i) is LBS.

We have X is BCS and i>>X=Y(i) for any i belongs to N.

We define LBS to be the smallest set which can satisfy these four claim with two map >> and S.

3.We define a function -> : N*LBS -> LBS

For any n belongs to N,

(1) n->φ=n

(2) n->X=(n+1)->(n>>X),for any LBS X≠φ.

4.We define a function KIRBN:N->N+

We define a LBS sequence A[n]:A[0]=[],A[n+1]=[][A(n)].

And KIRBN(n)=3->A(n).

KIRBN has growth rate Γ_0.

If the symbol comma can be used,this notation can have growth rate much more than ψ(I),but the definition is complex.

5.Some examples.

n>>[][[]][]=[][[]]

3>>[][[]]=[][][]

(Some space added to watch it more easily)

4>>3>>[] [ [] ] [ [] [] ]=4>>[] [ [] ] [ [] ] [ [] ]=[] [ [] ] [ [] ] [] [ [] ] [ [] ] [] [ [] ] [ [] ] [] [ [] ] [ [] ]

3>>[] [ [] [ [] ] ]=[] [ [] ] [ [] [] ] [ [] [] [] ]

3>>[] [ [] ] [ [] [] ] [ [] [] [] ]=[] [ [] ] [ [] [] ] [ [] [] ] [ [] [] ]

3>>4>>[] [ [] [ [] [ [] ] ] ]=3>>[] [ [] ] [ [] [ [] ] ] [ [] [ [] ] [ [] [] ] ] [ [] [ [] ] [ [] [] ] [ [] [] [] ] ]

=[] [ [] ] [ [] [ [] ] ] [ [] [ [] ] [ [] [] ] ] [ [] [ [] ] [ [] [] ] [ [] [] ] ] [ [] [ [] ] [ [] [] ] [ [] [] ] [ [] [] ] ]