It is tempting to just delete a wrong answer on mathoverflow.net. However, some errors are subtle, and it might be instructive to label some wrong answers as such, but leave them on MO to highlight pitfalls in reasoning about the problem at hand.

What do you think about this? Or should all wrong answers be deleted, no matter what?

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    $\begingroup$ Certainly mark them as incorrect. How about: let the one posting the erroneous answer choose whether to delete it? $\endgroup$ – Gerald Edgar Jun 22 '15 at 12:19
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    $\begingroup$ There are certainly precedents where a user leaves a wrong answer intentionally as a lesson to others coming to the question later. At the moment I'm having trouble coming up with a really good example of this. $\endgroup$ – Todd Trimble Jun 22 '15 at 13:34
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    $\begingroup$ If one likes to obtain badges, then one could obtain a peer "pressure badge" for downvoted incorrect answers or a "disciplined badge" for upvoted incorrect answers. I must admit that leaving an incorrect answer on MO has the advantage of showing the MO community where they have upvoted an answer in error and hence have upvoted the answer without reviewing the answer enough to determine (within reason) if it is correct or incorrect. In other words, leaving upvoted incorrect answers promotes the good MO practice of reviewing answers before upvoting. $\endgroup$ – Joseph Van Name Jun 22 '15 at 22:42
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    $\begingroup$ @Joseph, when you hover over the upvote arrow, it doesn't say, "This answer is correct." It says, "This answer is useful." $\endgroup$ – Gerry Myerson Jun 23 '15 at 0:27
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    $\begingroup$ This question appears to be a duplicate of this, but it seems inappropriate to close it given the nice answer Joonas has given here (which is not entirely redundant with the answers there). $\endgroup$ – Eric Wofsey Jun 24 '15 at 10:56
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    $\begingroup$ @Gerry Myerson. It says "this answer is useful" for all stack exchange communities, so the phrase "this answer is useful" represents the stackexchange network in general rather than the particular site mathoverflow. Of course, I should mention that if an incorrect answer has been labeled as incorrect I would not upvote it nor downvote it. I am sure that not many people would upvote an answer if they knew it was incorrect. $\endgroup$ – Joseph Van Name Jul 2 '15 at 3:46

I think many wrong answers should be kept for the reason you stated. But this only applies to answers that have mistakes that are interesting or easy to do; it is only worthwhile to keep answers if people can learn from them. Comments and downvotes can indicate that the answer is not good, and ideally it should be also indicated in the body of the answer as well. If an answer has a very negative score, the answerer may want to remove it to save their reputation (in any meaning of the word).

There are also answers that are not even wrong (to quote Pauli) — somehow nonsensical or unrelated to the question. Those should be removed.

If the answer is not nonsensical, I think it would be better to let the answerer make the deletion call, as Gerald Edgar suggested in the comments.

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    $\begingroup$ Here is an example of mine from MO. $\endgroup$ – Gerald Edgar Jun 23 '15 at 12:58

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