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I really don't understand your reasoning. I think it was a perfectly reasonable question.

If all sets were finite, how could the real numbers be defined?.

The all-sets-finite system is a possible axiom system, and I really think that it must have been studied. I remember hearing about it somewhere, but I just can't remember, and couldn't find in Google.

It is very clearly NOT a "too localized" question.

It is also NOT "subjective and argumentative" . If something is out of your area of expertise, it does not mean that it is subjective. If you could wait enough, maybe someone at least could give some references.

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    $\begingroup$ Harry, I don't think your comments of this type are helpful. They just sound a little arrogant to me. If I think a question is bad, I would downvote it, and maybe write a comment as to how it could be improved. $\endgroup$ – AgCl Jul 24 '10 at 12:52
  • $\begingroup$ @97832123: Please chill. I've cast the final vote for reopening, now that the question has been edited. $\endgroup$ – Akhil Mathew Jul 24 '10 at 13:39
  • $\begingroup$ Thanks for your vote Akhil! $\endgroup$ – AgCl Jul 24 '10 at 14:00
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    $\begingroup$ To clarify, I was telling AgCl that his question was awful. $\endgroup$ – 97832123 Aug 5 '10 at 10:08
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    $\begingroup$ I reviewed AgCl's original versions of the question, and they are much better than the "forced improvements". AgCl's comments are entirely correct, all of them: the closure was nonsense; the question was a good one (and the mathematical ideas in his original question were solid); the expertise-of-moderators issue is an important one here more than on MO. $\endgroup$ – T.. Aug 9 '10 at 6:19
  • $\begingroup$ @T..: No moderation was involved in closing or reopening this question, just votes of 5 normal users. $\endgroup$ – Larry Wang Aug 9 '10 at 16:28
  • $\begingroup$ The negative "improvements" were at the request of a moderator, as I recall. Expertise-of-users issues arise when wrongheaded statements about the posting elicit a chorus of amens, in this case in the form of votes to close. This is one reason expert moderators (e.g., solid MO level) are needed: to block the collective silliness of non-expert users. $\endgroup$ – T.. Aug 10 '10 at 3:27
  • $\begingroup$ Also, the revision history shows that 2 of the 5 user votes to close as "too localized" came from moderators. math.stackexchange.com/posts/501/revisions $\endgroup$ – T.. Aug 10 '10 at 3:34
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    $\begingroup$ @Sriva: Is it really a good idea to bump a year-old discussion to correct a typo? $\endgroup$ – Phira Oct 18 '11 at 15:25
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    $\begingroup$ @Phira Oops, sorry. (And thanks for pointing out!) I chanced upon this question through the recommendation of the site and I instinctively corrected the typo without paying attention to how old the question is. I will be more careful in the future. $\endgroup$ – Srivatsan Oct 18 '11 at 15:28
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    $\begingroup$ @Srivatsan I would have done the same as you, even though I realize it probably is best not to. I saw those typo's too, and well, it is like an impulse to edit sometimes. $\endgroup$ – Ellie Kesselman Dec 11 '11 at 19:49
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I answered the question. It's one that may sound "obviously" crazy but has a definite answer and has been studied and used in some very famous work in logic.

Premature closure is overzealous moderation. Let the questions sit a while, especially as it will take some time for the more knowledgeable mathematicians to come in from MO and elsewhere.

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  • $\begingroup$ Remember that the question has been edited substantially since it was originally asked. After it was changed to this form, the question was then reopened. $\endgroup$ – Larry Wang Aug 9 '10 at 6:34
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    $\begingroup$ The original form was clearer and better mathematically, and there was nothing wrong with it. $\endgroup$ – T.. Aug 9 '10 at 6:58
  • $\begingroup$ Thanks so much for the great answer again! I'll possibly add the original form back into the text. I'm really surprised to learn that finite ZF is equivalent to first order PA. $\endgroup$ – AgCl Aug 9 '10 at 14:01
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AgCl, I could not see how your question would make sense (as I explained in the comments). Finitism is inherently incompatible with the real numbers: you can't construct an uncountable set, more generally. Sure, you can specify, say, a real algebraic number, but these are only countable. More generally, cf. the countability of the computable numbers; these are the only ones I could imagine arising. For now, I think the question is also more vague than I would like.

Closure for these sorts of questions may be slightly harsh -- it will take some time to set a protocol. However, note that on MO, there have even been instances of moderators having their own questions closed; it is a fairly common occurrence. It should not be taken personally.

If I've erred and missed something important here, please explain, and I'll vote to reopen.

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    $\begingroup$ +1 for "It should not be taken personally." $\endgroup$ – Tom Stephens Jul 23 '10 at 1:40
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    $\begingroup$ Akhil, thanks for your reply. I agree that the question is not asked in the best possible way, but it is because the I only have a little knowledge about the concept I am asking about. So I tried to include all the details about what makes me ask this question, so that an expert would easily figure out what I wanted to learn. By the way, I have just realized that what I'm asking is closely linked to constructivism. I found this definition of a real number in wikipedia, which is close to what I was looking for: en.wikipedia.org/wiki/… $\endgroup$ – AgCl Jul 23 '10 at 2:26
  • $\begingroup$ @AgCl: The computable numbers are based on the intuitionistic continuum, and are not finitary. $\endgroup$ – Charles Stewart Jul 23 '10 at 8:56
  • $\begingroup$ +1 I was trying to make this point in my answer regarding the inherent incompatibility, but people didn't seem to be hearing. $\endgroup$ – Noldorin Jul 25 '10 at 11:18
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    $\begingroup$ Computable numbers are not based on the intuitionistic continuum. $\endgroup$ – T.. Aug 9 '10 at 8:12
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    $\begingroup$ @T..: Comments are no way to resolve this argument, so I have created a question, math.stackexchange.com/questions/1932/… $\endgroup$ – Charles Stewart Aug 9 '10 at 16:46
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On the scale between a question with a single, well-defined answer and a subjective discussion topic, I thought it was too far towards the latter. The title looks fine, but the first paragraph of your question reads "what would mathematics be like?" And you proceed to refer to 'all theory that has practical importance.'
Other distinct questions you asked were 'can we still construct the real numbers" and "how do we define e?" "How do we define sqrt(2)?" and "does anyone have references to where (all of the above, I suppose) have been published?" If you edited your question substantially to reduce the scope, and removed the vague language, I would support reopening.

Edit: To clarify, I don't think it's a bad question at all, just one appropriate for a discussion forum rather than a Q&A site.

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  • $\begingroup$ Thanks for the explanation, but I disagree. I believe someone who is qualified to answer my question would see that, all those questions are in the same direction and could be answered concretely in one or two paragraphs. Or s/he could give relevant references where I could find the answers. $\endgroup$ – AgCl Jul 23 '10 at 0:28
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    $\begingroup$ Although I don't know whether I am qualified to answer your question, I certainly understand it. Your questions are in roughly the same direction, but I think you really asked at least three distinct, but related questions, one of which is unanswerable. $\endgroup$ – Larry Wang Jul 23 '10 at 0:41
  • $\begingroup$ Kaestur, thanks anyway! I know it was not asked in the best possible way, but I still think it could have been answered in this site. $\endgroup$ – AgCl Jul 23 '10 at 2:37
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I have substantially edited the question, taking into account the critisisms. Please vote to reopen, if you think it is ok now.

One intersting thing to note about the question being closed is, 3 of the people who closed the question says it is "too localized", and 2 of them says it is "too vague", meaning it is "not localized enough"!

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  • $\begingroup$ I thought that was strange too, and I'm disappointed the others who voted to close didn't leave comments here. Anyways, your edited question looks good to me. Hope someone smarter than me can come up with a good answer! $\endgroup$ – Larry Wang Jul 24 '10 at 14:23

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