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I'm sick and tired of seeing those almost the same, but usually not quite duplicate questions asking something like

How is this set uncountable? We can put it in bijection with the natural numbers like this ...

Where common examples are $\mathcal P(\Bbb N)$ or infinite binary strings, or all the sequences of natural numbers, or whatever. And the usual mistake is that the bijection misses every infinite set, or any string with infinitely many $1$'s, or so on.

Let's make a canonical thread!

Great, let's start a canonical thread, but every time I try to write one, I get stumped. These questions are similar, but reducing the problem from one variant into another is for itself a legitimate question.

Is it going to be a good idea to start a big list thread for false proofs contradicting Cantor's theorem/diagonal? Should it be CW? Should it have some rules (e.g. one example per question, etc.)?

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  • 3
    $\begingroup$ Will this stop people from asking such questions? I think not. We need some smart software that says "Buddy! Did you just try to contradict Cantor? You're wrong, check this thread!" or something... $\endgroup$
    – Pedro Tamaroff Mod
    Feb 14 '15 at 22:38
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    $\begingroup$ @Pedro Isn't Asaf this software? ;) $\endgroup$
    – Lord_Farin
    Feb 14 '15 at 22:39
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    $\begingroup$ @Lord_Farin: Yes, that's me. And I want to optimize that software! $\endgroup$
    – Asaf Karagila Mod
    Feb 14 '15 at 22:48
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    $\begingroup$ Is the list countable? $\endgroup$
    – quid Mod
    Feb 14 '15 at 23:25
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    $\begingroup$ @quid: Whenever I try to make such list, there ends up being another one not on the list. So no. $\endgroup$
    – Asaf Karagila Mod
    Feb 15 '15 at 0:01
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    $\begingroup$ In general I consider abstract duplicates pretty useless. People tend to have very specific difficulties, and those who can use more general treatments to extricate themselves can usually find such treatments in their textbooks, lecture notes, etc. Your difficulty coming up with a canonical thread is pretty much a demonstration that this topic is especially ill-suited for an abstract duplicate. $\endgroup$
    – Brian M. Scott
    Feb 15 '15 at 22:24
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    $\begingroup$ @Brian: I think that 90% of the diagonal failures have the same answer, and it's almost always a one-liner, and it's almost always some infinite counterexample. "Isn't this an enumeration of $\mathcal P(\Bbb N)$?" - "No, it only enumerates finite sets, so $\Bbb N$ is not in its range"; or "Isn't this enumeration of all infinite binary strings?" -"No, the string "1111..." is not enumerated". And it goes on. This, instead, will encourage more than a few lines of answer. $\endgroup$
    – Asaf Karagila Mod
    Feb 15 '15 at 22:26
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    $\begingroup$ Even if somehow this is imposed, it will make an even more intangible community. New answerers will have to be taught before trying to answer. btw, anyone, can create and use such long listed thread outside MSE without preveting anyone else to give their own answers. $\endgroup$
    – Minimus Heximus
    Feb 15 '15 at 22:45
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    $\begingroup$ @user795571: How is it any different from me running around closing these as duplicates, or complaining about them being duplicates loudly in the comments? How is different from any other "frequently asked question" being closed as a duplicate? How is it different from the need to learn MathJax syntax and proper formatting? How is it any different from learning how to write a proper question or a reasonable answer? $\endgroup$
    – Asaf Karagila Mod
    Feb 15 '15 at 23:02
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    $\begingroup$ @Brian I'm sorry, but I fail to see why your assessment that "it's not worth the trouble" should prevent others, with different convictions, from doing the work? In general, I disapprove of such cynical considerations as "it's easier just to answer". Is there any cogent argument you wish to push forward, other than your gut feeling? $\endgroup$
    – Lord_Farin
    Feb 15 '15 at 23:03
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    $\begingroup$ @Lord_Farin: Your disapproval isn’t really relevant. The obvious fact is that if enough answerers either don’t know about it, find it easier just to answer the question than to dig up the thread, or simply want to try their hand at answering the question, be it for the practice or for the (very modest) ‘reputation’ that might accrue, then it will be of limited value. $\endgroup$
    – Brian M. Scott
    Feb 15 '15 at 23:18
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    $\begingroup$ I rememebr a programming forum in which each user was provided with a personal blog and they could link their blog posts as answers. But a hidden big-list question does not seem interesting and workable. $\endgroup$
    – Minimus Heximus
    Feb 16 '15 at 0:39
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    $\begingroup$ I downvoted because for similar reasons to Brian I don't think I would find such a resource useful. I wouldn't be actively sad if one was made anyway. $\endgroup$
    – Ben Millwood
    Feb 16 '15 at 22:53
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    $\begingroup$ It is worth trying as an experiment. Some possible issues: How does one determine the canonical problem? How does one find the canonical thread? Should folks be 'rewarded' for cleanup? In general, I think it is difficult to search (It is often more difficult to find an existing answer than to answer again) and many visitors are looking for a human guide rather than an answer. The latter is not the site intent, but I think it is the source of many questions. Peculiar as it may sound, the reward for me is often more the interaction (either with like minds or by way of enabling) than the answer. $\endgroup$
    – copper.hat
    Feb 22 '15 at 19:17
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    $\begingroup$ One other thing is that an answer is not just a function of the question, but also of the asker's skill set and 'mathematical maturity', so a static answer may not address many visitors' concerns. $\endgroup$
    – copper.hat
    Feb 22 '15 at 19:35
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I don't think an abstract duplicate for exactly those questions will be terribly useful.

For the class of questions Asaf describes, it's just as quick as complete to write

No, that doesn't work. You're missing all of the infinite whatsits.

as it would be to find and redirect to a duplicate, possibly have a dialogue with the asker about how, exactly, the duplicate is relevant to his question.

The distinguishing point here is that usually that one-line answer is the only thing the asker needs in order to see the error of his ways. In the vast number of cases they shut up sheepishly after that. So there wouldn't actually be any work saved by closing rather than just writing that answer another time.

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    $\begingroup$ I truly understand the sentiment of "one line answer is not worth a canonical thread with lots of answers". But this is also beneficial to the site, to have answers which explain these common mistakes. $\endgroup$
    – Asaf Karagila Mod
    Feb 20 '15 at 11:25
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Given that I have been routing on meta recently for a more prominent use of abstract duplicates, it will come as no surprise that I support this request.

However, as you already mention, there is the point of converting this idea to an actual workable solution. In this post, let me try to indicate some important considerations and principles that will hopefully help us to find the road to a successful implementation of this idea. I will indicate these in quote environments to highlight them.

First, to get some ambiguity expressed in the comment by Pedro out of the way:

An abstract duplicate for countability arguments will, as the name suggests, serve as an easy-to-find duplication target for common questions. It will not prevent the posting of new questions of this type.

It may help a bit against the latter (some people actually do search before posting), but it is in general not realistic to expect this.

A second, and in my opinion also quite important point:

The creation of abstract duplicates provides a means for the established community to provide a best-effort contribution with great, insightful answers. As such, it will be a valuable addition to the site from the perspective of creating a mathematical knowledge repository.

We spend a lot of time every day answering the same questions. I for one recognise the fatigue kicking in at times, meaning I don't take the time to properly elaborate on the underlying motifs. Abstract duplicates are a way to circumvent this.

There are also some issues with abstract duplicates that need to be addressed.

Abstract duplicates (ADs) have the risk of function creep: they will be used to close questions that are not quite addressed by the AD. We can see this type of situation in action on a daily basis with regards to the custom off-topic reason. It seems to be used a lot as a catch-all to close any question that somehow does not fit the site's (or, the user's) standards.

And indeed, even new questions that are supposed to be covered by the AD may not find their situation explained in appropriate detail. It is not realistic to expect a new AD to be optimal in the first run. We need to find a mechanism to facilitate improvement of the AD as time and insight progress.

An abstract duplicate (AD) has to find a way to deal with two particular types of dupe-closures:

  • Questions which are not supposed to be covered by the AD;
  • Questions for which the AD is not adequately equipped yet.

I think it depends on the actual AD at hand whether or not these will be big obstacles. An approach has to be agreed on for every AD, I would say, although it might be possible to come to a base agreement for ADs in general here on meta (but perhaps that would be a new thread).

Some users have indicated that they find the StackExchange model inadequate for generating new input on old material. This is a serious concern; particularly if one imagines revolutions in the approach of a subject (as e.g. algebraic geometry has gone through with the advent of category theory as a discipline of its own).

To address these concerns, I would say:

Questions of the type "How can the methods lined out in the AD be applied to this example?" can be very much on-topic. This provides a means for new users/experts to chip in. It may very well lead to adapting the AD if necessary (here lies an obvious connection with the previous point).

Conclusion

You may ask now: "All good and well, LF, but how to go from these generic deliberations to the case at hand?" A sensible question indeed, so I will address it now. Thanks for bearing with me so far.

For the countability AD, I envisage the following:

  • The question itself could ask something like "What are common fallacies in (dis)proving a set is countable, and how can they be fixed?"
  • Each answer ought to go through a specific argument, explain where it goes wrong, and if applicable, present a corrected argument that actually proves uncountability.
  • Each answer also ought to point to specific instances of this argument. This can be done either by a simple list (like "$\Bbb R$ with this-and-this purported bijection") or by linking to duplicates that employ the technique.

I think that in particular the third point is important. It shows how the argument is actually encountered in practice, establishing important ties from the necessarily abstract context of an AD to the specific context of a real-life example. This can also work really well with the "explain the AD" type of question I mentioned above — after such questions have been adequately answered, they can be closed and added as further examples to enhance both the visibility and the quality of the AD.

Well, that ended up a bit longer than I anticipated, but I hope it is a valuable basis for discussion. As always, comments and additions are welcome.

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  • $\begingroup$ Good, thanks for writing this. The way I see it, it's a thread with an abstract question, and a list of links to the answers (where the link details the particular set being discussed there, or something like that). Each answer will include a discussion over one fake proof; since those are usually short enough to be written into an answer. It will also serve to cover future instances not quite covered by those already in the list, simply by adding a good answer there whenever it is needed. $\endgroup$
    – Asaf Karagila Mod
    Feb 15 '15 at 21:32
  • $\begingroup$ (And if it gets all the way to 25+ answers, then maybe we can stop, and reconsider this altogether.) $\endgroup$
    – Asaf Karagila Mod
    Feb 15 '15 at 21:32
  • $\begingroup$ @Asaf Since many of the arguments are quite similar for different sets, I see possibilities for grouping multiple of these into one answer. This can mitigate explosion of the number of answers needed. $\endgroup$
    – Lord_Farin
    Feb 15 '15 at 21:37
  • $\begingroup$ Well, the question is, where do you stop the abstraction. All those arguments are essentially the same. Look closely at the supposed enumeration and use the diagonal argument to produce a counterexample. Most often it's an example with infinitely many $1$'s or whatever; or an infinite set; or an infinite co-infinite set; etc. All those are really the same. But reducing from one to the other is something I wouldn't necessarily close if someone would ask about. So my point is that those need to be addressed separately, even if the idea is close. $\endgroup$
    – Asaf Karagila Mod
    Feb 15 '15 at 21:42
  • $\begingroup$ @Asaf I guess you're right. We'll find out the balance as we go, I suppose. $\endgroup$
    – Lord_Farin
    Feb 15 '15 at 21:43
  • $\begingroup$ Do you have a good title in mind for a thread like that? $\endgroup$
    – Asaf Karagila Mod
    Feb 15 '15 at 21:55
  • $\begingroup$ @Asaf In line with what I wrote, perhaps "Common fallacies in countability proofs, and their resolution"? $\endgroup$
    – Lord_Farin
    Feb 15 '15 at 21:57
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    $\begingroup$ How horribly functional. :-) $\endgroup$
    – Asaf Karagila Mod
    Feb 15 '15 at 22:00

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