User blog:Ikosarakt1/Extension of Chris Bird's array notation.

In this blog post, I will write about my extension of Chris Bird's array notation.

Define Bird's hierarchy:

\(B_{\alpha}(n) = \lbrace n,n [X] 2 \rbrace\), where \(\alpha\) specifies a separator and vice versa.

Notice that Bird's hierarchy grows faster even than fast-growing, for example:

\(f_{\varepsilon_0+1}(n) \approx \lbrace n,n,2 [1 \ 2] 2 \rbrace\) \(B_{\varepsilon_0+1}(n) \approx \lbrace n,n [2 \ 2] 2 \rbrace\), since \([2 \ 2]\) has level \(varepsilon_0+1\)

About separators with level below \(\varepsilon_{\Omega+1}\) (i.e. Bachmann-Howard ordinal) and the gist of array notation you can read here.

In 7 part of Bird's stuff, he defines "generalized" backslash separator: \(\_n\). There are good, but what about \(\_1,2\)? This separator outranks all separators that was defined by Bird. I define:

\(\lbrace n,n [1 \_1,2 2] 2 \rbrace = H(n)\), this function was defined by Bird at the end of paper.

Now I present list of separators beyond this. Let \(\rightarrow\) is shorthand for "has level".

\([2 \ 1,2 2] \rightarrow \varepsilon_{\Omega+1}\)