User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-32697988-20180127233730/@comment-30118230-20180129201022

It's not a fixed point but rather a supremum.

A function $$F(\alpha)=\alpha+1$$ has NO fixed points for any ordinal,however a function $$F(\alpha)=1+\alpha$$ DOES have fixed points,the first one of which is $$\omega$$.

$$C(0,\alpha)$$ however in not defined as $$1+\alpha$$ but rather $$\alpha+1$$,that's why it has not fixed points.

That's why $$C(1,0)=\sup\{C(0,0),C(0,C(0,0)),C(0,C(0,C(0,0))),....\}$$ and $$C(1,\alpha)$$ is the $$1+\alpha$$th such supremum.