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?

I'm studyng De Rham theorem for smooth manifolds and I noticed a thing. Many authors prove the theorem for a convex subset of $\mathbb{R}^n$ and then they show that the theorem is true for any open subset of $\mathbb{R}^n$. Defined a De Rham basis for the manifold $M$ as a basis $\mathcal{B}$ suc...

I was reading Bruhat Tits' "Groupes réductifs sur un corps local : I. Données radicielles valuées", available here. Let $(G,B,N,S)$ be a Tits system and $\phi : G \to G^{'}$ be a group homomorphism. It is called $B-$adapted (resp. $BN-$ adapted) if $\ker (\phi) \subset B$, and for every $g \in...