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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 bracketstring):
(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…
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1.we define a set BCS (bracketcomma string):
(1)any nonnegitive integer is in BCS
(2)if a_1 and a_2 and ... and a_n is in BCS,then the string a_1,a_2,...,a_n is in BCS
(3)if a_1 and a_2 and ... and a_(n+1) and x_1 and x_2 and ... and x_n is in BCS,then the string a_1[x_1]a_2[x_2...[x_n]a_(n+1) is in BCS for any integer n2.we recursive define an order on BCS:
for BCS x and y
(1)if x has "outside" comma more than y,then x>y
(2)if both have the same number of "outside" comma,let x=u,v y=s,t if u>s then x>y if u=s and v>t then x>y
(3)if both don't have outside comma,write x as the from in 1.(3),let the biggest x_i be u,the biggest x_i of y be v.if v is empty and u is not then x>y. if u>v then x>y. if u=v , write x=a[u]b , y=c[u]d , both a and c do…
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