User blog comment:Dchew89/An Ordinal Describing the Growth Rate of my d(n) Function/@comment-30754445-20190218043613/@comment-30754445-20190218234235

It can certainly be done.

The main question is whether this saves you any amount work, when compared to the ordinary approach. You still need to manually input all the fundamental sequences of the ordinals below your Δ0. And if already know how to do that, you don't really need the Δ0-concept in the first place.

One thing you can do, however, is to keep the definition of Δ0 open-ended. You can use this approach to create an always-developing fundamental sequence of (say) ωck1. It is a work that, in principle, can never be completed. But you can continue to make it stronger and stronger, one item at a time, as you master the understanding of larger and larger ordinals.