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Formalising Foundations in Dependent Type Theories

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See  Vladimir Voevodsky - What if Current Foundations of Mathematics are Inconsistent .  Subscribe to  Formalization Seminar Cambridge . Subscribe to ItaLean Conference .  The state of the art three years ago: Subscribe to  Lean Prover Community .  I came here because I heard this talk by Simon Willerton last week and he seems to be able to interpret some ideas like extanaturality from higher category theory into 2-category theory, but I am not sure about that: he doesn't say that's what he's done, it's just how it seems to me.  Subscribe to  Topos Institute .