User:Googleaarex/Dimensional Array Ordinals

Up to \(\omega\)
Simple.

x(1)y(2)z... = x+y+z...

Up to \(\omega^\omega\)
0(1)1 = \(\omega\)

x(1)y = \(\omega y + x\)

0(2)1 = \(\omega^2\)

x(1)y(2)z = \(\omega^2 z + \omega y + x\)

x(1)y(2)z... = \(... + \omega^2 z + \omega y + x\)

Up to \(\omega^{\omega^{\omega}}\)
0(1)0(1)1 = \(\omega^2\)

x(1)y(1)z... = \(... + \omega^2 z + \omega y + x\)

0(2)1 = \(\omega^\omega\)

0(2)0(1)1 = \(\omega^{\omega + 1}\)

0(2)0(2)1 = \(\omega^{\omega 2}\)

0(3)1 = \(\omega^{\omega^2}\)

0(n+1)1 = \(\omega^{\omega^n}\)

Up to \(\varepsilon_0\)
0[(1)(1)]1 = \(\omega^\omega\)

0[(1)...(1)]1 (n+1 (1)'s) = \(\omega^{\omega^n}\)

0[[(2)](1)]1 = \(\omega^{\omega^\omega}\)

0[[(2)](1)(1)]1 = \(\omega^{\omega^{\omega 2}}\)

0[[(2)][(2)](1)]1 = \(\omega^{\omega^{\omega^2}}\)

0[[(2)(2)](1)]1 = \(\omega^{\omega^{\omega^\omega}}\)

0[[(2)...(2)](1)]1 (n+1 (2)'s) = \(\omega^{\omega^{\omega^{\omega^n}}}\)

0[[[(3)](2)](1)]1 = \(\omega^{\omega^{\omega^{\omega^\omega}}}\)

0[[[[(4)](3)](2)](1)]1 = \(\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}\)