User blog:Kite in rainbow/A weak and clear version of my Bracket-Comma notation ver2,it reaches Γ(0)

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 b2.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,...

<p style="font-weight:normal;font-size:16px;line-height:26px;">(4)Let X is an element of BS,X≠φ,n=n(X),every X(i) is LBS.

<p style="font-weight:normal;font-size:16px;line-height:26px;">If 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)).

<p style="font-weight:normal;font-size:16px;line-height:26px;">Define m to be the biggest number s.t. m>>X(n) belongs to S(X(i)).

<p style="font-weight:normal;font-size:16px;line-height:26px;">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.

<p style="font-weight:normal;font-size:16px;line-height:26px;">If for any i belongs to N we have Y(i) is LBS,then X is BCS and i>>X=Y(i) for any i belongs to N.

<p style="font-weight:normal;font-size:16px;line-height:26px;">We define LBS to be the smallest set which can satisfy these four claim with two map >> and S.

<p style="font-weight:normal;font-size:16px;line-height:26px;">3.We define a function -> : N*LBS -> LBS

<p style="font-weight:normal;font-size:16px;line-height:26px;">For any n belongs to N,

<p style="font-weight:normal;font-size:16px;line-height:26px;">(1) n->φ=n

<p style="font-weight:normal;font-size:16px;line-height:26px;">(2) n->X=(n+1)->(n>>X),for any LBS X≠φ.

<p style="font-weight:normal;font-size:16px;line-height:26px;">4.We define a function KIRBN:N->N+

<p style="font-weight:normal;font-size:16px;line-height:26px;">We define a LBS sequence A[n]:A[0]=[],A[n+1]=[][A(n)].

<p style="font-weight:normal;font-size:16px;line-height:26px;">And KIRBN(n)=1->A(n).

<p style="font-weight:normal;font-size:16px;line-height:26px;">KIRBN has growth rate Γ_0.

<p style="font-weight:normal;font-size:16px;line-height:26px;">If the symbol comma can be used,this notation can have growth rate much more than ψ(I),but the definition is complex.

<p style="font-weight:normal;font-size:16px;line-height:26px;">5.Some examples.

<p style="font-weight:normal;font-size:16px;line-height:26px;">n>>[][]=[]

<p style="font-weight:normal;font-size:16px;line-height:26px;">3>>[]=[][][]

<p style="font-weight:normal;font-size:16px;line-height:26px;">(Some space added to watch it more easily)

<p style="font-weight:normal;font-size:16px;line-height:26px;">4>>3>>[] [ [] ] [ [] [] ]=4>>[] [ [] ] [ [] ] [ [] ]=[] [ [] ] [ [] ] [] [ [] ] [ [] ] [] [ [] ] [ [] ] [] [ [] ] [ [] ]

<p style="font-weight:normal;font-size:16px;line-height:26px;">3>>[] [ [] [ [] ] ]=[]  [ [] ]  [ [] [] ]  [ [] [] [] ]

<p style="font-weight:normal;font-size:16px;line-height:26px;">3>>[] [ [] ] [ [] [] ] [ [] [] [] ]=[] [ [] ] [ [] [] ] [ [] [] ] [ [] [] ]

<p style="font-weight:normal;font-size:16px;line-height:26px;">3>>4>>[] [ [] [ [] [ [] ] ] ]=3>>[]   [ [] ]   [ []  [ [] ] ]   [ []  [ [] ]  [ [] [] ] ]    [ []  [ [] ]  [ [] [] ]  [ [] [] [] ] ]

<p style="font-weight:normal;font-size:16px;line-height:26px;">= []   [ [] ]   [ []  [ [] ] ]   [ []  [ [] ]  [ [] [] ] ]    [ []  [ [] ]  [ [] [] ]    [ [] [] ]  ]    [ []  [ [] ]  [ [] [] ]    [ [] [] ]    [ [] [] ]  ]