User blog comment:Emlightened/Ordinals in Type Theory/@comment-11227630-20180128144447/@comment-27513631-20180128205732

× adds nothing in that sense.

Nat has strength \(\varepsilon_0\), as it corresponds to the dialecta translation of PA.

Ord requires Nat by definition of Ord, and very likely has strength \(\vartheta(\varepsilon_{\Omega+1})\).

Adding an Ord operator to the theory, with parameter replacing where Nat occurs in Ord has strength \(\vartheta(\Omega_\omega\).

Adding dependent types (I believe dependent product and a single universe will do) should have strength \(\vartheta(\Omega_{I+\omega})\).

If we were to extend this to full inductive-recursive types, then we would have \(\vartheta(\Omega_{M+\omega})\), however, I believe it would be more conducive to googologist's interests to instead have a (large?) type Ord and non-positive ways of making new 'regular cardinals'.