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This is about question 38468, which asks whether one can make sense of such things as 1/aleph_0 and other seemingly nonsensical quantities. As Richard Borcherds commented, one can make sense of such quantities in the context of surreal numbers. It seems to me that this is an interesting answer to the question and it seems to invalidate closure as "not a real question."
As a moderator, my vote to reopen the question would automatically reopen the question. This is a boundary case where I wouldn't do that without community support, so I'm starting this discussion to poll the community. You may consider this to be a single vote to reopen.
I'm going to play devil's advocate here...
Robin: "Is there any reseatch [sic] on this?" seems like a legitimate reference request to me.
Ryan: Yes, Richard Borcherds gave the best answer I can think of. Having an answer in a comment has never been a valid reason to close. I also don't agree with your wikipedia test: what could the OP have looked for to land on that page? [Edit: Ryan and Mariano have given very good answers to this question.]
François, he would have to google "infinitely small numbers" (the very title of the question!) and read through the first result, which is the Wikipedia page on Infinitesimals, which lists number systems which do have such numbers.
This makes for a third-millenium variant of Jeopardy: instead of having to phrase responses as questions, you'd have to phrase them as paths in the Web Graph...
Ironically, question 38468 is slowly creeping up as a top hit when googling "infinitely small numbers." It would be unfortunate for a poor soul to click on such a dead end...
Jonas, there was a proposal for vote trading in this thread. Nothing definitive came out of this discussion. Right now, my policy is to only cast fifth votes unless there is a very good reason to do anything different.
François: Having an answer in a comment has never been a valid reason to close.
This is something that I feel needs clarification, because it is not an uncommon situation. My attitude is that if the question is so elementary or so wrong that a short comment suffices to completely answer it then it would be appropriate to close the question. In other words, frequently the reason why the question was answered in a comment may is that the question is inappropriate for MO, with the act of answering in a comment serving as evidence. Sometimes, transferring the comment to an answer leads to a quick resolution (the OP simply accepts the answer and the question goes away); sometimes the OP is not mathematically qualified to understand the answer or even to formulate the question properly, which leads to unproductive discussions of little benefit to anyone else. If the OP is satisfied with the response and the respondent does not wish to repost his or her comment as an answer, I usually vote to close as no longer relevant.
Closing a question because it has been answered in comments is generally inappropriate. The point is that closing a question prevents any new answers, which is contrary to MO's objectives.
That said, when a question is "so elementary or so wrong" then there surely are other reasons to close the question. In this case, the question should be closed for those reasons. It is perfectly fine to comment on a closed question. I find that the best comments indicate how to ask a better question, but answering a closed question in a comment is not necessarily bad behavior.
One thing to keep in mind is that we all want to encourage people to ask good questions on MO. Comments and votes to close are best used to steer users (especially new users) in the right direction. In this particular case, the right direction might be math.SE, but I'm not sure about that and nobody has suggested that in a comment.
Let me explain my vote to close.
MO is for research level questions: i.e., questions that graduate students or post-PhD mathematicians encounter in the course of their research, or those of equivalent level and content. But no matter how I looked at the question, I couldn't see it as research level. First, the question as it has been asked seems to be based on the assumption that cardinal numbers have reciprocals (which are also cardinal numbers). But this is a profound misunderstanding of a kind that a professional mathematician or student of mathematics would not make. Second, Professor Borcherds has pointed out that there is some number system in which it makes sense to take such reciprocals: the surreal numbers. That's an insightful remark, but did the OP have this in mind? If s/he had expressed problems with .9999999... = 1, then would we answer the question by reference to non-standard analysis?
For me, the clincher here is this: suppose the question does refer to the surreal numbers. I personally know next to nothing about this topic, but I am told (by wikipedia) that they form an ordered Field (the capital letter is hinting that they are a proper class, not a set). But of course I know that in any ordered field, if 0 < x < y, then 0 < 1/y < 1/x. In other words, even charitably interpreting the question in a domain where it makes sense, the answer is immediate.
If anyone can explain to me why there's more to the question than this, I will be happy to vote to reopen.
I agree that the question is poorly worded. I personally put the emphasis on the end: Is there any research on this? In any case, I just edited the question to make it easier to read. My reading may be generous, but I thought the question was whether one could make any sense of 1/aleph_0 other than "0" or "undefined."
Alex, have you read Pete Clark's explanation of his vote to close before complaining? I think that the penultimate paragraph makes a good case for closing as "not a real question"; perhaps, the paragraph before it does as well. Given what you know about the closing system, namely, that the reasons for closing are immutable, every vote to close must specify exactly one reason, and the majority vote decides which reason is displayed, why keep beating the dead horse? If your problem is with the act of closing itself, which is certainly a legitimate position, then simply say so.
OK. Working out the answer I would give to the question and comparing with Pete's motivation, I am now convinced that this question should remain closed. I will post a comment asking the OP to transfer the question to math.SE where he will surely get some interesting answers.
@markvs: I think you're confusing surreal numbers with hyperreal numbers.
alex_o, it truly is impossible because we don't have access to the source. I have seen this explained to you multiple times on meta, but for some reason, you keep ignoring these explanations.
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