User blog comment:Deedlit11/Ordinal Notations III: Collapsing Higher Cardinalities/@comment-11227630-20160501075609

In the "$$\theta$$ function up to $$\theta(\Omega_{\Omega_{\Omega...}},0)$$" section, $$C_0(\alpha,0)$$ is empty, then 0 is not in $$C(\alpha,0)$$, so $$\theta(\alpha,0)=0$$. Well, my question is, why it's not $$C_0(\alpha,\beta)=\beta\cup\{0\}$$? Since previous versions of $$\theta$$ function do this.

And what does $$\Omega_0$$ mean? Is it 0, or $$\omega$$?