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Revision of Induction from Fri, 02/01/2009 - 00:20

When this article is written, it should have examples of several different styles of inductive proof: the usual kind where you deduce P(k) from P(k-1), the slightly more sophisticated kind where you deduce it from the fact that P(j) is true for every j<k, induction over more sophisticated well-ordered sets, use of the well-ordering principle, etc.

Transfinite induction is discussed in a separate article.

Parent article

Proving universal statements

Comments

Moved the discussion to the comments

When this article is written, it should have examples of several different styles of inductive proof: the usual kind where you deduce P(k) from P(k-1), the slightly more sophisticated kind where you deduce it from the fact that P(j) is true for every j<k, induction over more sophisticated well-ordered sets, use of the well-ordering principle, etc.

Induction without an indexing set

"The generalized induction is harder because we don't have a nice indexing set like the natural numbers. Is there a way of doing induction without the crutch of an indexing set? Yes there is: you look for a minimal counterexample."

The above is from this article by Prof Gowers: http://www.dpmms.cam.ac.uk/~wtg10/bounded.html.

I feel that this could be put somewhere, but I am not sure where. The connection between the well-ordering principle and induction could also be mentioned (albeit briefly, since it doesn't seem like a trick).