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).
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.