# 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"). Nov 2, 2017 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. Nov 2, 2017 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 Mod
Nov 2, 2017 at 7:11
• @GerryMyerson That's about how I go about estimating which tag applies too. Nov 2, 2017 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 Mod
Nov 2, 2017 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 Mod
Nov 2, 2017 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! Nov 2, 2017 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 Mod
Nov 2, 2017 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. Nov 2, 2017 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. ;) Nov 3, 2017 at 4:55
• Well, I've been a Dr. for less than a day, so you didn't miss by much.
– Asaf Karagila Mod
Nov 3, 2017 at 7:03
• Congratulations doctor!!! I know that you are not really fond on congratulations but I have no mercy now. Nov 3, 2017 at 10:13

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.)