After reading the latest 'make the FGH work with ordinals' posts, I came up with one that, as far as I know, is the only one that doesn't stop at \(\Gamma_0\) or a similar sized ordinal.

Let \(\Omega\) be an uncountable regular ordinal. We define fundamental sequences for all ordinals of cofinality \(\Omega\) that are \(<\varepsilon_{\Omega+1}\):

\(\Omega^\alpha[\beta] = \Omega^{\alpha[\beta]}\) if \(\alpha\) is a limit ordinal.

\(\Omega^{\alpha+1}[\beta] = \Omega^\alpha\cdot\beta\)

\((\alpha+\gamma)[\beta] = \alpha+\gamma[\beta]\) if \(\alpha\) is additively principal and \(\alpha\cdot\omega>\gamma\)

All non-zero ordinals \(<\varepsilon_{\Omega+1}\) can be represented as \(\mu\) or \(\mu[\beta]\), where \(\mu\) has cofinality \(\Omega\) and \(0<\beta<\Omega\).

Now, define the functions \(H_\alpha:\Omega\mapsto\Omega\) for each \(\alpha\leq\varepsilon_{\Omega+1}\):

\(H_0(\beta) = \beta\)

\(H_{\alpha+1}(\beta) = H_\alpha(\beta+1)\)

\(H_{\mu[\alpha]}(\beta) = \sup\{H_{\mu[\gamma]}(\beta):\gamma<\alpha\}\) (\(0<\alpha<\Omega\))

\(H_{\varepsilon_{\Omega+1}}(\beta) = \sup\{H_{\Omega\uparrow\uparrow n}(\beta):n<\omega\}\)

This can generate large ordinals, for instance: \(H_{\varepsilon_{\Omega+1}}(\omega) = \text{BHO}\)

As the fundamental sequences up to the BHO are still rather complex, I haven't created a normal ordinal hierarchy to go with the large countable ordinals

Lastly, a note: This seems to me to be a way that the idea of an ordinal collapsing function can be introduced, without immediately introducing any new concepts (except possibly uncountable ordinals), as the reader will likely already be familiar with the Hardy and Fast-Growing Hierarchies. Hence, if anyone would like to use this blog post in the 'Introduction to Ordinal Collapsing Functions' page (we have one of those, right?), you can cite this page.