# Is this elementary-set-theory or set-theory

I've been involved with this question, in which the presence of the tag set-theory has been contested (check out the edit history). The asker is dealing with questions about equinumerousness of infinite sets, as well as the Cantor-Berstein Theorem. From what I can glean from the tool tips, it's a little ambiguous.

In my own experience as a tutor, I've certainly seen set theory with a more elementary treatment, and one that is more along the lines of the description in the elementary-set-theory tag (focusing on basic operations, Venn diagrams, etc). The set-theory description includes studying "large cardinals" (a term which I'm not certain if I'm correctly interpreting), but certainly looking indirectly at these infinite cardinalities could be construed as set-theory.

Could someone with more experience in these matters please weigh in on this?

• The removal of the tag on the question was done by someone with a lot of experience in this. From the description of the tag, this particular question does not seem to fit (large cardinals is something more precise in set theory than just "comparing uncountable sets"). – Tobias Kildetoft Nov 2 '17 at 5:25
• My impression is that there's a gap between what's covered by the (description of the) set-theory tag and what's covered by the (description of the) elementary-set-theory tag, and this question falls into that gap. But I'd be inclined to go along with elementary for this question, as it does seem like the sort of thing that would be done in an introductory, not an advanced, course. My rule-of-thumb for set theory is, if I can understand it, it's elementary. – Gerry Myerson Nov 2 '17 at 6:28
• There is no ambiguity, this is elementary. And as @Tobias says, large cardinal is a technical term and not just uncountable sets. – Asaf Karagila Nov 2 '17 at 7:11
• @GerryMyerson That's about how I go about estimating which tag applies too. – Tobias Kildetoft Nov 2 '17 at 7:21
• @Gerry: I can't use that rule of thumb... :( mine is "would this have been covered in that intro course we had in BGU", which is sometimes too high but nonetheless works fine for most part. – Asaf Karagila Nov 2 '17 at 11:13
• Let me also say that while I disagree with the idea that this question is suitable to the set-theory tag, I do welcome this meta question and the fact that some people are actively and in a civilized manner try to participate in the tagging process by trying to understand what fits which tags! – Asaf Karagila Nov 2 '17 at 11:37
• Hey, @Asaf, what took you so long? We had to wait two whole hours for you to show up in this discussion! – Gerry Myerson Nov 2 '17 at 11:58
• @Gerry: I count 4.5 hours, but I'm a set theorist, so I'm probably wrong about counting finite things. And I had other, slightly more urgent things to do, like finally get some sleep (e.g. between my comment and my answer). – Asaf Karagila Nov 2 '17 at 12:11
• @Asaf, it currently says the question was asked 7 hours ago, and your first comment came 5 hours ago, so I'll stick with 2 hours. But sleep is good. – Gerry Myerson Nov 2 '17 at 12:21

This post is without a doubt an elementary set theory.

While the topics of a basic course in set theory differ between universities (some have a lot of advanced materials, while others might not even have a set theory course on its own), the rule of thumb is that if this is something someone who is not a set theorist should be able to understand and give a rough answer, then it's probably elementary.

Let me also add that large cardinals are a technical topic in set theory which is rather advanced. Wikipedia has some information for you, in case you're interested.

Finally, let me say that the rule of thumb is just a rule of thumb. There were times that I disagreed with retagging from one set theory tag to another, and if I felt this is a gray area case, I would consider voicing my opinion or undo (or edit further). We play a lot of this by ear, but as far as I see it, if Andrés removed the set theory tag, then it's very unlikely that there is a reason for the tag to be there.

• Apparently, I missed that you're a "Dr." already. My belated well-wishes. ;) – J. M. is not a mathematician Nov 3 '17 at 4:55
• Well, I've been a Dr. for less than a day, so you didn't miss by much. – Asaf Karagila Nov 3 '17 at 7:03
• Congratulations doctor!!! I know that you are not really fond on congratulations but I have no mercy now. – drhab Nov 3 '17 at 10:13

You mentioned that you looked at the tooltip shown when adding and (i.e., the tag-excerpt). It is worth mentioning that you can also view the full tag-info for a tag. Although I'd agree that in the case of elementary-set-theory and set-theory it does not add too much.

Another useful things when choosing tags if you are not sure is to look at similar questions. In this case, you could look at related questions shown in the sidebar. And now in the comments several questions about the same - or very similar - thing were mentioned. (The difference is that the linked question is a question.) You can see that they are tagged . For some general advice on how to choose tags, see: How am I supposed to use tags?

One aspect in which the tags and are special is that there are two users who follow this tag very closely and help maintaining it - among other things, by retagging posts which are tagged incorrectly. They are Asaf Karagila and Andrés E. Caicedo. So AFAICT they are the most experienced users on the site regarding tagging set-theoretical questions. (Still, I had an occasional disagreement with one of them about these tags or some other tags related to set theory. I am only saying that their past activity means that they do have authority in questions related to these two tags. So if a retagging was done by one of them, it is usually in accordance with the way these tags were used in the past.)

About this specific question, I'd say that the sentence "focusing on material usually covered in undergraduate set theory texts" in the tag-info covers basic cardinal arithmetic, but the tag-info could probably be more specific about this. In the past cardinal arithmetic was explicitly mentioned in the tag-info, it was later edited away. Cardinal arithmetic still can cover wide range of topics from calculating $\aleph_0^{\aleph_0}$ (which I would definitely consider elementary-set-theory) to, for example, Bukovský-Hechler theorem, where I would probably consider (set-theory). (It seems that questions about this result are tagged elementary-set-theory.)

In connection with this particular question it is probably worth mentioning that there also is a separate tag named .