User:Wythagoras/Dollar function/Nested Array Notation

Nested Array Notation is the fifth part of Dollar Function.

Formal definition
\(\bullet\) can be anything

\(\circ\) is a group of brackets.

\(\diamond\) is a group of zeroes and serperators.

1. If there is nothing after the $, the array is solved. The value of the array is the number before the $.

2. \(a\$b\bullet=(a+b)\$\bullet\)

3. \(a\$\circ[0]\bullet\circ=a\$\circ a\bullet\circ\)

4. \(a\$\circ[\bullet+1]_c\bullet\circ=a\$\circ[\bullet]_c[\bullet]_c...[\bullet]_c[\bullet]_c\bullet\circ\) with a \(\bullet\)'s

5. If the bracket contains a zero and the bracket has other content, you can remove the zero.

6. If the active bracket has level k and a zero in it, search for the least nested bracket with level (k-1) with the same array in it, nest that bracket a times in the place of the level k bracket and put [0] on the place where the active bracket was.

7. \( b\bullet,c = [[0,c-1]_{[b-1\bullet,c][b-1\bullet,c]}]\)

8. \( b\bullet,\diamond,c,\bullet = [[\bullet,\diamond,[\bullet\diamond,c-1,\bullet]_{[b-1\bullet,\diamond,c,\bullet][b-1\bullet,\diamond,c,\bullet]},c-1,\bullet]\)

9. \([0,\diamond,c,\bullet] = [[0],\diamond,c-1,\bullet]\)

10. \(a\$ b\bullet(c\bullet)d\bullet =\)

\(a\$[e\{\underbrace{[0](c-1\bullet)[0]...[0](c-1\bullet)[0]}_{\bullet}(c-1\bullet)\underbrace{[0](c-1\bullet)[0]...[0](c-1\bullet)[0]}_b(c\bullet)d-1\bullet\}]\) 11. \([b\bullet(\diamond,0,c,\bullet)1] = [0(\diamond,[b-1\bullet(\diamond,0,c,\bullet)1][b-1\bullet(\diamond,0,c,\bullet)1],c-1,\bullet)[0]]\)

12. \([b\bullet(\diamond,0,c,\bullet)d] = [0(\diamond,[b-1\bullet(\diamond,0,c,\bullet)d-1],c-1,\bullet)[0](\diamond,0,c,\bullet)d-1]\) 13. \([0(\diamond,0,c,\bullet)1] = [[0](\diamond,0,c-1,\bullet)1]\)

14. \([0(\diamond,0,c,\bullet)d] = [[0](\diamond,0,c,\bullet)d-1]\)

S1. The outermost bracket is always level 1

S2. If there is no bracket with level (k-1), add it directly after the level k bracket.

S3. Zeroes at the and of the array must be removed

S4. (0) is shorthand for a comma

S5. \([\diamond(a)0(b)\bullet] = [\diamond(b)\bullet]\) if a<b

The active bracket is the bracket with the lowest level. The brackets can be ordered by level in FGH, and then removing outermost bracket. Or you look to: smallest bracket type, least number of nestings, smallest number inside the bracket.

Analysis
\(0](0,1)[0\) is the 1st ordinal in the second \(\omega^\omega\) space

\(0](1,1)[0\) is the 2nd ordinal in the second \(\omega^\omega\) space

\([[0](1,1)0,1]\) is the 3rd ordinal in the second \(\omega^\omega\) space

\(0](1,1)[0](1,1)[0\) is the 1st ordinal in the third \(\omega^\omega\) space

\(0](2,1)[0\) is the 2nd ordinal in the second \(\omega^{\omega+1}\) space

\(0](1,2)[0\) is the 1st ordinal in the second \(\omega^{\omega2}\) space

\(0](1,[0])[0\) is the 1st ordinal in the second \(\omega^{\omega^2}\) space

\(0](1,0,1)[0\) is the 1st ordinal in the second \(\omega^{\omega^2+1}\) space

\(0](0,1,1)[0\) is the 1st ordinal in the second \(\omega^{\omega^2+\omega}\) space

\(0](0,0,2)[0\) is the 1st ordinal in the second \(\omega^{\omega^22}\) space

\(0](0,0,0,1)[0\) is the 1st ordinal in the second \(\omega^{\omega^3}\) space

\(0]([0](1)1)[0\) is the 1st ordinal in the second \(\omega^{\omega^\omega}\) space

\(0]([0](1)[0])[0\) is the 1st ordinal in the second \(\omega^{\omega^{\omega+1}}\) space

\(0]([0](3)[0])[0\) is the 1st ordinal in the second \(\omega^{\omega^{\omega^2}}\) space

\(0]([0]([0])[0])[0\) is the 1st ordinal in the second \(\omega^{\omega^{\omega^\omega}}\) space

Limit is the \(\varepsilon_0\) space, which is also the strength of Hyp cos' Pseudo Nested Array Notation.