User blog comment:Deedlit11/Ordinal notations II: Up to the Bachmann-Howard ordinal/@comment-11227630-20140830020542

I don't think the $$\vartheta$$ is a bijection. For example, there's no $$\alpha$$ that fits $$\vartheta(\alpha)=1$$, since all the $$\vartheta$$-ordinals are epsilon ordinals.