User:Wythagoras/Dollar function/Hyper Nested Array Notation

Hyper-Nested Array Notation is the sixth part of Dollar Function.

Introduction
We start with \(0]<0>1]\), which is \([[0]([0]([0](...)[0])[0])[0\) with [0] nested layers. We can have larger things, but, it is interesting when we get larger and larger arrays, until \(0]<0>[0]<0>[0]...[0]<0>[0]<0>[0\) What is it? It is \([[0](1)<0>1]\), not \([[0]<1>1]\). That makes this notation powerful.

Additional rules
\(\triangle\) is a <> structure.

15. \([b\bullet<0>\triangle1] = [[0]([b-1\bullet<0>\triangle1][b-1\bullet<0>\triangle1])\triangle1]

16. \([0<0>\triangle1] = [[0]\triangle1]\)

17. \([0<0>\triangle d] = [[0]<0>\triangle d-1]\)

The other rules work in <> brackets just like in [] brackets, example: <1> = <0><0>...<0><0>.

Analysis
\([0]<0>1\) is a \(\varepsilon_0\) structure

\([0](1)<0>[0]\) is a \(\varepsilon_0\omega\) structure

\([0]([0])<0>[0]\) is a \(\varepsilon_0\omega^\omega\) structure

\([0]([0]<0>1)<0>1\) is a \(\varepsilon_0^2\) structure

\([[[0]<0><0>1]\) is a \(\varepsilon_1\) structure

\(0<1>1\) is a \(\varepsilon_\omega\) structure.

\([0]<0><1>1\) is a \(\varepsilon_{\omega+1}\) structure

\(0<1><1>1\) is a \(\varepsilon_{\omega2}\) structure

\(0<2>1\) is a \(\varepsilon_{\omega^2}\) structure

\(0<[0]>1\) is a \(\varepsilon_{\omega^\omega}\) structure

\(0<[[0]_2]>1\) is a \(\varepsilon_{\varepsilon_0}\) structure

\(0<[[0]_2[0]_2]>1\) is a \(\zeta_0\) structure

\(0<[[0]_3]>1\) is a \(\vartheta(\varepsilon_{\Omega+1})\) structure

\(0< [[0,1 >1]]\) is a \(\psi(\psi_I(0))\) structure

\(0<[[[0]<0>1>1]]\) is a \(\varepsilon_0\) structure structure. If we say that the \(X\) notation maps structures, it is equal to \(X(X(\varepsilon_0))\)

The ordinal that describes \([[0]<[0]_2>1]\) is the first ordinal \(\alpha\) such that \(\alpha\) equals its own structure. It is also \(X(\Omega)\)

\(0<[0,1]>1\) has level \(X(\psi_I(0))\)

etc.