User:Wythagoras/Ordinals/Strong ordinal notation

This is my ordinal notation.

Part 1
\(C_0(0)_{\alpha}[0] = 0\)

\(C_{\alpha}(\beta+1)[0] = C_{\alpha}(\beta)+1\)

\(C_{\alpha}(\beta)[n+1] = \omega^{C_{\alpha}(\beta)[n]}\)

Part 2
\(C_{\alpha}(0)_{\beta+1}[n+1] = C_{\alpha}(C_{\alpha}(0)_{\beta+1}[n]+1)_{\beta}\)