User blog comment:Emlightened/Intro/@comment-1605058-20160109180510/@comment-5529393-20160110020117

Well, we could for example encode the Buchholz hydra into a function by encoding a labelled rooted tree into a natural number: Encode a root tree with label n as $$2^n$$, and encode a tree with label n and children $$T_1, T_2, \ldots, T_m$$ as $$2^n p_1^{F(T_1)} p_2^{F(T_2)} \ldots p_m^{F(T_m)}$$, where $$ p_i$$ is the ith prime and $$F(T_i)$$ is the code for $$T_i$$. It seems straightforward then to describe the Buchholz Hydra process in terms of these codes, but I don't feel like doing that at the moment.

Of course, the same could be done for any notation based on arrays or structures of any kind. I guess what we are looking for is something more original/simple that can't be easily seen to be a structure in code.