User blog:Googleaarex/FGH Notation

Here a notation about FGH.

Number Notation
Target: \(\omega\)

@ can be Array.

Reaching \(\omega\)
FGH(a,0) = a+1

FGH(a+b,0) = a+(b+1)

FGH(@,0) = @+1

FGH(1,@-1) = FGH(1,@) = 1

FGH(FGH(2,0),0) = FGH(2,1) = 4

FGH(@,@,n+1) = FGH(FGH(@,@,n),n-1)

FGH(@,@,1) = FGH(@,@-1)

FGH(n,m) = FGH(n,m,n)

Rules of Number Notation
1. FGH(Any,0,Any) = Any+1

2. FGH(n,n@) = FGH(n,n@,n)

3. FGH(Any,n+1@,1) = FGH(Any,n@)

4. FGH(Any,n+1@,m+1) = FGH(FGH(Any,n+1@,m),n@)

Bracket Notation
Target: \(\varepsilon_0\)

@2 can only [] brackets that even number of brackets

Reaching \(\varepsilon_0\)
FGH(a,[1]@) = FGH(a,a)

FGH(a,b+1@) = FGH(FGH(...FGH(FGH(a,b@),b@)...,b@),b@) (a nested)

FGH(a,[1]@) = FGH(a,a@)

FGH(a,[b+1]@) = FGH(a,[b][b]...[b][b]@) (a b's)

[@+1](a) = [@][@]...[@][@] (a @'s)

Rules of Bracket Notation
5. FGH(n,@2[1]@2) = FGH(n,@2n@2)

6. FGH(n,@2[m+1@]@2) = FGH(n,@2[m@][m@]...[m@][m@]@2) (n [m@]'s)

7. FGH(Any,@20@2) = FGH(Any,@2@2)

Array-Type Notation
Target: \(\Gamma_0\)

Reaching \(\Gamma_0\)
FGH(a,[...[2@]...]) = FGH(a,[2@])

FGH(a,[20]) = FGH(a,... [[1 ...]]) (a nested [] brackets)

FGH(a,[1[2@]]) = FGH(a,[2@][2@]...[2@][2@]) (a [2@]'s)

FGH(a,1[2@]) = FGH(a,2@][2@]...[2@][2@) (a [2@]'s)

FGH(a,[2@+1]) = FGH(a,...[[1[2b]...]]) (a nested type 1 [] brackets)

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