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

A piece of feedback - although I don't know how well this applies to other readers: Your analysis is too detailed to be easy to read. You spend over 30 lines to achieve a mere \(\omega^2\), which makes reading less of a task of understanding and more a task of scrolling.

I'm not asking you to simplify your analysis - the thoroughness is arguably commendable, but you would save both yourself and your readers much effort by, after a pattern repeating some small number of times (say 4), expressing it in its general form. I don't know if this is what you personally want to do, honestly, as it's entirely reasonable for you to replicate what you find most understandable, but a simple

C(0,0) = 1

C(0,1) = 2

C(0,2) = 3

C(0,a) = a+1

C(1,0) = ω

C(0,C(1,0)) = ω+1

C(1,1) = ω2

C(1,2) = ω3

C(1,a) = a+ω

Is less informative, true, but slightly easier to read and also more general.

Besides, if you want to go at a rate where you can plausibly entirely define the notation, you're going to have to cut corners somewhere eventually (it's admirable that you haven't yet), and it's best to do it without sacrificing any rigour.