User blog:Ynought/Gamma function redefined

So my 2nd last blog post was a ordinal function.And it failed,pretty badly.So i looked into ordinal functions a bit more and i came up with this:

\(C_0(\alpha,\beta)=\alpha\cup\beta\cup\{0,1,\omega,\Omega\}\)

\(C_{i+1}(\alpha,\beta)=\{\delta,\delta+\eta,\omega^\delta,\delta^\eta,\gamma(\mu)|\delta,\eta,\mu\in C_i(\alpha,\beta);\mu<\alpha\}\)

\(C(\alpha,\beta)=\{0,1,\omega,C_{n<\omega}(\alpha,\beta)\}\)

\(\gamma_\beta (\alpha)=\text{min}(\delta|\gamma_{\omicron<\beta}(\xi<\alpha)<\delta<\Omega,\delta\in C(\alpha,\beta),\forall k;f_\delta(k)\leqslant g_\delta(k+1))\)

As far as i can tell this should work,but i'm not certain,so please feel free to correct me

(don't expect answers today,since im gonne for the rest of the day)