a$b = a+b
a$[0] = a*2
a$[0][0] = a*2^2
a$[1] = a*2^a ~ a^a has arrow-level \(\uparrow\)
a$[1][1] ~ (a^a)^(a^a) = a^a^{a+1}
a$[2] has arrow-level \(\uparrow\uparrow\)
| Dollar Function without $ | Arrow Level |
|---|---|
| [1] | \(\uparrow\) |
| [n] | \(\uparrow^n\) |
| [[0]] | \(\uparrow_1\) |
| [n[0]] | \(\uparrow_1\uparrow^n\) |
| [[0][0]] | \(\uparrow_1\uparrow_1\) |
| [[1]] | \(\uparrow_2\) |
| [[[0]]] | \(\uparrow_{\uparrow_1}\) |
| [[0]_2] | \(\uparrow_{,\uparrow}\) |
| [[1]_2] | \(\uparrow_{\uparrow,\uparrow}\) |
| [[[0]_2]_2] | \(\uparrow_{\uparrow_{,\uparrow},\uparrow}\) |
| [[0]_3] | \(\uparrow_{,\uparrow\uparrow}\) |
| [[0]_[0]_2] | \(\uparrow_{,\uparrow_{,\uparrow}}\) |
| [[[0],1]] | \(\uparrow_{,,\uparrow}\) |
| [[[1],1]] | \(\uparrow_{\uparrow_{,,\uparrow},,\uparrow}\) |
| [[[1],1]] | \(\uparrow_{\uparrow_1,,\uparrow}\) |
| [[[[0],1],1]] | \(\uparrow_{\uparrow_{,\uparrow_{,,\uparrow}},,\uparrow}\) |
| [[[0,1]_2,1]] | \(\uparrow_{\uparrow_{,,\uparrow},,\uparrow}\) |
| [[[0,1]_3,1]] | \(\uparrow_{,\uparrow,\uparrow}\) |
| [[[0],2]] | \(\uparrow_{,,\uparrow\uparrow}\) |
| [[0,[[0],1]]] | \(\uparrow_{,,\uparrow_{,\uparrow_{,,\uparrow}}}\) |
| [[0,[0,1]_2]] | \(\uparrow_{,,,\uparrow}\) |