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Kaba
Posts:
289
Registered:
5/23/11


Induction beyond finities
Posted:
Mar 5, 2013 6:15 PM


Hi,
Is it possible to somehow generalize induction so onepieceatatime proofs would generalize to all sets?
Here's an example:
Let V be a nondegenerate symmetric bilinear space over F. I have managed to prove that if V is finitedimensional, 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)?
 http://kaba.hilvi.org



