Coq: Prop versus Set in Type(n)

Question

I want to consider the following three (related?) Coq definitions.

Inductive nat1: Prop :=
  | z1 : nat1
  | s1 : nat1 -> nat1.

Inductive nat2 : Set := 
  |          


        

Answer 1

Type : Type is inconsistent.

Impredicative Set with excluded middle implies proof irrelevance, so impredicative Set with proof relevance, e.g. true <> false, refutes excluded middle, which intuitionism isn't supposed to do.

Therefore we leave impredicativity in Prop and the rest of the type hierarchy gives us predicativity.

By the way,

forall P : nat1 -> Prop, P z1 -> (forall n : nat1, P n -> P (s1 n)) -> forall n : nat1, P n

is provable. Don't ask me what's the benefit of Coq only automatically proving that other weaker induction principle...

Also, have you read this chapter of CPDT?