Is it possible to derive induction for the church-encoded Nat?

Question

I was just wondering if it is possible to derive induction for the church-encoded Nat type on Idris, Agda, Coq and similar. Notice this is a different issue from doing it on

Answer 1

I think there is no formal proof that it's impossible, but generally expected that it can't be done. See e.g. the introduction to this paper by Aaron Stump.