The proof theoretic ordinal of Martin-Löf type theory is $\Gamma_0$.
What about HoTT and what about other flavors of type theory (the ones related to the lambda cube for instance)?
2026-03-25 01:28:00.1774402080
What is the proof theoretic ordinal of Homotopy Type theory?
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I believe this is unknown: current ordinal analysis only gets up to around $\Pi^1_2$-$\mathsf{CA_0}$, and my recollection is that HoTT is vastly stronger than that - around the level of ZFC + two inaccessibles. It's almost impossible to describe how vast the gulf between subsystems of second-order arithmetic and ZFC is; I would be profoundly surprised if the proof-theoretic ordinal of anything near ZFC was computed in my lifetime.
As to much weaker type theories, these slides of Buchholtz and this paper of Setzer have some relevant information. But again, unless my understanding is wildly off-base any variant of HoTT with around the same power is going to be far out of reach of current ordinal analysis.