User blog comment:Emlightened/Ordinals in Type Theory/@comment-11227630-20180128144447/@comment-30118230-20180128220938

Which vartheta function are you using?

I don't know any defined beyond $$\psi(\Omega_\omega)$$

If you want to use functions beyond that,then use some variant of $$\psi$$,and if you want to enumerate ordinals beyond the TFB ordinal,then you have to use Rathjen's $$\psi$$ or some variant of it. If you want to go beyond Rathjen's ordinal,which you seem to imply to since you use ordinal collapsing functions which take in cardinals of the sort $$ \Omega_{M+\omega} $$,then you have to use some extention of Rathjen's psi function or create/point out your specific choice of notation first. Otherwise,it's a bit ambiguous.

Anyways,that was a little nit-picky.

If what you said is true,then Typed Lambda Calculus with Nat is as strong as ACA0; with Nat by definition of Ord is as strong as KP; with "Ord  operator to the theory, with parameter replacing where Nat occurs in Ord" is as strong as $$\Pi^1_1-CA_0$$; with dependend types it's as strong as Martin-Löf type theories with W-types and one universe; with full inductive-recursive types is as strong as KPM+ ect.