User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-27513631-20180129204203/@comment-27513631-20180129222822

I mean like? Technically, that would be cutting corners, as you're not stating the general case. However, you're also going far beyond the point where it's reasonable to assume 'laypeople familiar with ordinals can easily guess what this provided term is, assuming the system is "well behaved" (informally speaking)', so you're definitely not in that sense.

Also, as the terms are well-ordered (isomorphic to an ordinal), there's no ambiguity in using shorthands throughout, like C(1,0) = ω, C(Ω,0) = ε0, possibly C(α,β) = ωα+β later in the analysis.