User blog comment:PsiCubed2/My attempt for creating an ψ(ψᵢ(0))-level notation for ordinals/@comment-24920136-20170329070238/@comment-27513631-20170329194547

"I mean, how on earth are we supposed to visualize an ordinal like ψ(ψᵢ(0))?" I'd say the same, in principle, for even \(\varepsilon_0\). We can't view these ordinals directly, at least without great effort. What we (and most people who observe them) are interested in is the structure, and that is arguably considerably easier to understand, and even visualise. We can see \(\varepsilon_0\) as a tree, \(\psi(\Omega_\omega)\) as a tree with vertices labelled with integers, and \(\psi(\psi_I(0))\) as a tree with vertices labelled with trees, with intricate laws specifying the order of the trees.

Fundamental sequences for ordinals with countable cofinality generated by a '\(C\)-like' function like this are almost always trivial, by the way, if you don't restrict yourself to the sequences having nice arithmetic properties.

As far as how the pair sequence notation works? I think one of the easier ways is by analogy: compare the pair sequence notation to the Buchholz Hydra (with finite labels) in the same way you'd compare Beklemishev's Worms to the Kirby-Paris Hydra. There are a lot of similarities, and the Hydras are more easily open to interpretation as ordinals.