The following is written in the first chapter of Shoenfeld's text on mathematical logic.
"...suppose that a collection C is defined by a generalized inductive definition. Then in order to prove that every object in C has property P, it suffices to prove that the objects having property P satisfy the laws of the definition. Such a proof is called a proof by induction on objects in C".
Am I wrong to think that this is mistaken? Wouldn't this be merely showing that objects with property P belong to C, rather than showing that objects in C have property P?