User blog:Fejfo/Array notation arrays as ordinals

As I was reading about array notations, I realised that most could be thought of as a combination of:
 * A base: for refilling numbers
 * A prime: for iteration
 * An "ordinal array': this names the functions and diagonises by "collapsing" the array with the value of the array with the prime decreased by 1.

So you could define: And to formalise: Then: And a noation like would be equivalent to linear arrays ( a serves as both the base and the prime).
 * {a...}[n]={a-1...}
 * {0…0,p...}[n]={0…n,p-1...}
 * \({...}=sup{n<ω : {...}[n] }\)
 * \({a1,a2,…,aN}=ω^n ⋅ aN + … + ω⋅a2 + a1\)
 * A{0,...}=2                   (1 doesn't generate enough nesting)
 * A{a,0...}=a+2             (+2 to be consistent with the previous rule)
 * A{a,...}=A{a,...}[A{a-1,...}]

Ofcourse the notation could be easily expaned to tetrational level: Which would give a notation upto \(ε_0\)
 * , = {0...0} = the ordinal 0
 * {… 0 σ p…}[n] = {… n σ[n] n σ[n] … σ[n] n σ p-1...} with σ>0 and n seperators σ[n]

Is this a new idea? It can't be, but I didn't find any information about it. Since I've heard some array notations go beyond OCF's now I think this could be a usefull tool to define large ordinals.