Why Church-encoded types aren't sufficient to express inductive proofs?

Question

I've heard some claims that the calculus of constructions without inductive types isn't powerful enough to express proofs by induction. Is that correct? If so, why isn't the Church-encoding sufficient for that?

Answer 1

How would you prove inside the pure CoC that the induction principle holds of the Church numerals? See Thomas Streicher's, Independence of the induction principle and the axiom of choice in the pure calculus of constructions.

Answer 2

There is a result recently published within Annals of Pure and Applied Logic in which church encoded data are realizers of their own induction principle. In this system, the induction principle for natural numbers, trees, lists... are derivable. The core calculus doesn't have any datatype constructors packaged in. It starts at an extrinsic (curry style) System F adding 3 typing constructs: implicit product, heterogeneous equality, and dependent intersection.

Geuvers result does not apply to this calculus and is mentioned in the paper.

paper: "From Realizability to Induction via Dependent Intersection", by Aaron Stump