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Topic: Induction beyond finities
Replies: 1   Last Post: Mar 5, 2013 6:46 PM

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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 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)?

--
http://kaba.hilvi.org

Date Subject Author
3/5/13 Kaba
3/5/13 Shmuel (Seymour J.) Metz