By $MLTT + M$ I mean basic Martin Lof type theory with general $M$-types. It well known that the proof-theoretic ordinal of $MLW$ (MLTT with W-types) is the Bachmann-Howard ordinal, but is the proof theoretical ordinal of the analogous theory $MLTT + M$ known?
2026-03-25 07:48:31.1774424911
Has the proof theoretic ordinal of $MLTT + M$ been studied?
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According to this article, in $MLTT$ $M$-types may actually be constructed from $W$-types, so the answer is: Probably not, and simply because adding $M$-types to the language does not add any power to $MLTT$, since $M$-types are already expressible in $MLW$.
It would be an interesting question to see how far this generalizes to other types of inductive an coinductive constructions, see my question here for something along those lines, but from the recentness of this article, this is probably somewhat of an open/active research project.