When is proof by contradiction necessary?

Great post on Gower’s blog on when contradiction is appropriate. The first comment by Tao lends additional considerations to the post with a reference to non self-defeating objects!

Gowers's Weblog

It’s been a while since I have written a post in the “somewhat philosophical” category, which is where I put questions like “How can one statement be stronger than an another, equivalent, statement?” This post is about a question that I’ve intended for a long time to sort out in my mind but have found much harder than I expected. It seems to be possible to classify theorems into three types: ones where it would be ridiculous to use contradiction, ones where there are equally sensible proofs using contradiction or not using contradiction, and ones where contradiction seems forced. But what is it that puts a theorem into one of these three categories?

This is a question that arises when I am teaching somebody who comes up with a proof like this. “Suppose that the sequence $latex (a_n)$ is not convergent. Then … a few lines of calculation … which…

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