User:Wythagoras/Dollar function/Hyper Nested Array Notation II

Hyper Nested Array Notation II is the seventh part of Dollar Function. Hyper Nested Array Notation I was only the start of the hyper-nested arrays. The second part uses arrays inside the <> brackets, and then we get the same nesting as with the normal (0(0(...)1)1) brackets, giving us a <0/2> bracket, and, finally a <0/0/0...0/0/1> bracket, which is the limit of Hyper-Nested Array Notation II.

The arrays work the same as in Nested Array Notation.

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

16. \([0<0>\triangle1] = [[0]([[0]([[0]([[0](...)\triangle1]1)\triangle1]1)\triangle1]1)\triangle1]\)

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

18. \([0<\diamond/0/k\bullet>\triangle1] = [[0]<\diamond/[0]<\diamond/[0]<\diamond/.../k-1\bullet>\triangle1/k-1\bullet>\triangle1/k-1\bullet>\triangle1]\)

19. \([b\bullet<\diamond/0/k\bullet>\triangle1] = [[0]<\diamond/[b-1\bullet<\diamond/0/k\bullet>\triangle1][b-1\bullet<\diamond/0/k\bullet>\triangle1]/k\bullet>\triangle1]\)

Analysis
\(0<0,1>1\) has level \(\psi(X(Q))\)

\([0](1)<0,1>1\) has level \(\psi(X(Q\omega))\)

\(0<0><0,1>1\) has level \(\psi(X(\varepsilon_{Q+1}))\)

\(0<1><0,1>1\) has level \(\psi(X(\varepsilon_{Q+\omega}))\)

\(0<[0]_2><0,1>1\) has level \(\psi(X(\varepsilon_{Q+\Omega}))\)

\(0<0,1><0,1>1\) has level \(\psi(X(\varepsilon_{Q2}))\)

\(0<1,1>1\) has level \(\psi(X(\varepsilon_{Q\omega}))\)

\(0<[0]_2,1>1\) has level \(\psi(X(\varepsilon_{Q\Omega}))\)

\(0<0,2>1\) has level \(\psi(X(\varepsilon_{Q^2}))\)

\(0<[0](1)1>1\) has level \(\psi(X(\varepsilon_{Q^\omega}))\)

...

\(0<0,1/2>1\) has level \(\psi(X(Q_2))\)

\(0<0,1/[0]>1\) has level \(\psi(X(Q_\omega))\)

\(0<0,1/0,1>1\) has level \(\psi(X(Q_Q))\)

The limit is \(\psi(X(\alpha \mapsto Q_\alpha))\)