User blog comment:Dchew89/General Idea for my d(n) Function/@comment-35470197-20190306060453/@comment-35470197-20190306062827

It is not sufficient for you to fix fundamental sequences for \(\omega\) and \(\varepsilon_0\), because \(\omega^2\) can be equipped with the following fundamental sequence so that \(d(4) = \omega^2\): \begin{eqnarray*} 0,1,\omega,\omega+1,\omega \times 2, \omega \times 2+1, \omega \times 3,\ldots \end{eqnarray*} What you need to do is to fix the system of fundamental sequences to all ordinals, where the fundamental sequence for a general ordinal \(\alpha\) means a strictly increasing sequence of the cardinality \(\textrm{Cof}(\alpha)\). Otherwise, \(d(n)\) is completely ill-defined.