Is it possible to somehow generalize induction so one-piece-at-a-time proofs would generalize to all sets?
Here's an example:
Let V be a non-degenerate symmetric bilinear space over F. I have managed to prove that if V is finite-dimensional, then V has an orthogonal basis. This proof uses induction: the idea is to take one vector at a time, and reduce to smaller problems.
But such a theorem seems disappointing, since for vector spaces I can prove that every vector space has a basis, by assuming the axiom of choice (well, Zorn's lemma). I would like to prove that every such V has an orthogonal basis (I don't actually know whether that is true or not). Can the above proof be saved by some generalized induction, or does a possible proof have to be modified to a form where I can apply Zorn's lemma (or equivalent)?