User blog comment:Bubby3/Taranovsky's C 1st system introduction and rules./@comment-30118230-20180131194820

I guess you could define $$C(\alpha+1,\beta)$$ as the supremum of $$C(\alpha,\beta),C(\alpha,C(\alpha,\beta)),C(\alpha,C(\alpha,C(\alpha,\beta))),.....$$ although I would define it as the $$1+\beta$$th supremum to the sequence $$C(\alpha,0),C(\alpha,C(\alpha,0)),C(\alpha,C(\alpha,C(\alpha,0))),....$$.