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…Read more >
1.we define a set BCS (bracket-comma 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 n
2.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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