Induction on Church-encoded natural numbers (which I will call indNat) can not be defined within the Calculus of Constructions.
If we assumed indNat as an axiom, is there an untyped term that would have the semantics of the induction function? What if we had Scott-encoded natural numbers and fix?
Can induction be inhabited in CoC + fix (such that the function would have the proper semantics, not an infinite loop)?
Yes, there is such an untyped function, and it turns out that it is equivalent to the untyped erasure of iteration for church numerals.
The Cedille project has been doing lots to give types to these functions, and the core concept there is the dependent intersection, which provides a limited form of self reference that is enough to derive induction principles. This paper will probably interest you.
If you have $fix$ you can definitely implement induction principles, using structural recursion on the natural number. This is how they do it in CiC and Coq, with an extra restriction that the recursive calls must be on a structurally smaller value. Without this condition, you can write induction, but you also can write nonterminating terms with the same type as an induction principle.
