User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180130182451

All your analysis after $$\omega^{\varepsilon_0+1}$$ is wrong.

$$C(\alpha+1,\beta)=\sup\{C(\alpha,\beta),C(\alpha,C(\alpha,\beta)),C(\alpha,C(\alpha,C(\alpha,\beta))),.....\}$$ and is only valid if $$\alpha\leq \beta$$.

In this case $$a=b=C(\Omega,0)=\varepsilon_0$$ so the ordinal for $$\varepsilon_0\omega$$ is $$C(C(\Omega,0)+1,C(\Omega,0))$$