I just noticed a question that is obviously wrong here, and the OP doesn't seem to acknowledge it when somebody pointed out. (To clarify, the question was ok, but the OP seemed convinced that his polynomial was symmetric. My choice of example may not be so good, but I have seen other wrong questions in the past.) I voted to close it, and my choice was "unclear what you're asking." However, I feel it isn't really my reason, and other choices don't seem to correspond to my reason to close the topic. What is the most appropriate reason for closing in this case?
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3 Answers
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If you vote to close the standard reason is "unclear." As "it’s hard to tell exactly what [OP is] asking," in the sense that one cannot know what type of answer they are expecting when they insist something wrong needs to be shown. (Moreover, in the old days, one would have closed it as "not a real question" and Unclear is the successor to NARQ.) If you feel your concern is not sufficiently well-documented with the close reason, you can write a comment with further explanation.
Or, you close with a custom reason, that is "off-topic" then "other." In this case you get a free form text-field where you can give whatever reasoning you like, and this reason will be posted as a comment right away. An advantage of this over writing a usual comment is that these reasons are collected in a central list. This can be useful to find out which new standard close reason might be needed.
(Or still differently, you do not vote to close at all and explain in a comment or in an answer why what is postulated in the question is not true. In the specific case, I think I may have opted for this.)
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The following is just my opinion. Others might say something different.
A couple of things
There is the issue of whether the question should be closed for lack of context. Some will say that since the only context/attempt provided by the OP is that they used a computer program indicating that the polynomial isn't symmetric. I might personally be ok with leaving the question open.
I don't understand your classification of the question as being a "wrong question". The OP might indeed be confused about symmetric polynomials and whether the polynomial is symmetric. Therefore the question isn't "wrong" and there is good possibilities for providing an answer that could be helpful to the OP. If you see a question that is "wrong" in the sense that it doesn't make sense, I would close it as "unclear what you are asking".
In summary, I would leave the question open and provide an answer as to whether the polynomials is symmetric or not (like what the current answer does).
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Often the most difficult part of responding to a Question is helping the OP to clarify it, particularly when the problem amounts to some kind of misunderstanding.
Notwithstanding the difficulty and frustration of these situations (both for the OP and the Reader), these are certainly opportunities for learning (aka teaching moments). I think labeling these as "wrong questions" is not the best approach, but I appreciate the reasoning behind that.
Here's an example I've been wrestling with:
https://math.stackexchange.com/questions/1842792/g%c3%b6del-number-and-recursive-enumerable-set
The OP did respond to my original request for clarification, though not in a fully satisfactory way. So I'm trying again. Eventually I will tire, of course, if things continue to go in circles.
At that point I would probably vote to close as unclear what you're asking rather than for lack of context. My thinking about this distinction is that the OP has responded to requests for clarification, showing the minimal investment in the Question that counts (for me) as context. It just will not have resulted (assuming the worst) in a Question that makes much sense.
Potentially one or more Readers will post answers that patiently explain why the Question makes no sense (i.e. nothing can be said about incompleteness or about recursive enumerabilty unless more is known about "system $E$" than that it concerns "some standard arithmetic"). My comments will have laid some groundwork for that sort of answer (although I'll probably lack the enthusiasm to post one myself).
