User blog comment:Emlightened/Ordinals in Type Theory/@comment-11227630-20180128144447/@comment-27513631-20180128223801

I believe Deedlit has a \(\vartheta\) defined up to \vartheta(\Omega_{\Omega_{\Omega_\cdots}})\); I just fall into the habit of using \(\vartheta\) due to \(\psi(\alpha\)\) being ambiguous for many \(\alpha<\varepsilon_{\Omega_\omega+1})\).

And perhaps, but there are definitely OCFs which define it, and afaik they all coincide at that point (\(\psi(\Omega_{M+\omega})\).

And well ye, dependent products and a universe and Ord are a subtheory of MLTT1W, and my reasoning for their strength is that it should be a pto-maximising fragment; heuristically speaking, there's no reason that \(\Sigma\) or \(=\) will ever be needed in a sequence of maximising functions, and afaik all the W-types can be encoded from the special case Ord and extensional \(=\).