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I was about to retag this question when I noticed that neither of the or tags seems to really want questions of this tag.

The tag wikis say:

elementary-set-theory

This tag is for elementary questions on set theory, spanning topics usually found in introductory courses in set theory, in addition to review sections of graduate textbooks in the same field. Topics include intersections and unions, differences and complements, De Morgan's laws, Venn diagrams, relations, etc. More advanced topics should use (set-theory) instead.

set-theory

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

The question I was looking at was about the most elementary level of understanding of how transfinite ordinals and cardinals behave and relate to each other.

This is clearly more advanced than the example topics listed for -- which seems to cover mostly set algebra inside $\mathcal P(U)$ for some appropriate $U$, and in general only topics that are commonly necessary to use and understand the set-based mathematical jargon of standard undergraduate topics outside set theory itself.

On the other hand, it also seems to be clearly less advanced than the topics listed for . Most of those topics are only taught to students who are particularly interested in set theory, logic, or foundations.

I'd argue there's a small group of "intermediate" topics, comprising things like

  • infinite cardinals and ordinals, countability, axioms of ZFC

which are distinguished by not merely being "jargon" that mathematicians use to structure non-set-theoretic ideas -- they are "real" set theory, so to say -- but on the other hand they're also often taught as "general education" for students of other areas of mathematics, and are something a "well-rounded" mathematician, even outside set theory, needs to have some basic understanding of.

Of these topics, "axioms of ZFC" are in list, but that's not really a clear indication of where cardinals and ordinals belong.

I would propose to add "infinite cardinals and ordinals" to the list, but it's also conceivable that we should really move them to instead, possibly together with basic questions about how to use the ZFC axioms.

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  • $\begingroup$ The standard applications of Zorn's lemma: existence of (Hamel) bases, maximal ideals, algebraic closures, Tychonoff,... might also fall into this gap. But such questions may be better served without either tag. After all, we learn about them in topical courses (algebra, topology or functional analysis). $\endgroup$ – Jyrki Lahtonen Aug 12 at 3:37
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    $\begingroup$ (cont'd) Which leads to a convenient litmus test (for me): If I can understand it, it's elementary. $\endgroup$ – Jyrki Lahtonen Aug 12 at 3:41
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    $\begingroup$ I will just point out that revision 4 of the [tag-info for elementary-set-theory] contained "transfinite induction, well-orders, ordinal and cardinal arithmetic". It was removed in revision 5. See also this comment exchange around the time when the tag-info was edited. $\endgroup$ – Martin Sleziak Aug 12 at 4:38
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    $\begingroup$ I will try to weigh in tomorrow. By the way. I did not miss this thread. $\endgroup$ – Asaf Karagila Aug 12 at 22:46
  • $\begingroup$ I vote for using the set-theory tag. The mention in the existing wikia there of "transfinite hierarchies" seems sufficient, even if the particular Question is susceptible of an "elementary" explanation. $\endgroup$ – hardmath Aug 13 at 14:15
  • $\begingroup$ FWIW I have pinged Andrés E. Caicedo to comment here. @Asaf is already aware of this thread. IIRC Asaf has removed set-theory from some questions where transfinite induction played a (not necessarily significant) role. Let's wait. $\endgroup$ – Jyrki Lahtonen Aug 15 at 8:11
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    $\begingroup$ I'm not a set theorist, but I think a reasonable distinction is that applied set theory (i.e. set theory that is applied to other areas of mathematics) should fall under elementary-set-theory, because in some sense usage of set theory is about understanding the exact axioms and knowing how to invoke them in the right way to obtain useful tools (such as transfinite induction or recursion, or the well-ordering theorem) for non-set-theoretic mathematics. In contrast, the study of set theory itself should fall under set-theory. $\endgroup$ – user21820 Aug 18 at 2:27
  • $\begingroup$ @user21820 That could be a useful way of describing the difference. It feels like it is also close to the current practice. $\endgroup$ – Jyrki Lahtonen Aug 18 at 18:21
  • $\begingroup$ @AsafKaragila: Want to weigh in? =) $\endgroup$ – user21820 Aug 19 at 11:47
  • $\begingroup$ @user21820: Yes. But to be honest, I've found myself on the border between being fairly busy and being a bit burnt out on meta issues (and frankly, in general) recently, which is why I did not write anything on this thread yet. $\endgroup$ – Asaf Karagila Aug 19 at 16:48
  • $\begingroup$ @AsafKaragila: Oh okay thanks! Take a break from SE! =) $\endgroup$ – user21820 Aug 19 at 17:03
  • $\begingroup$ @user21820: Let's not get carried away. SE is my way to procrastinate, not think about mathematics, and at the same time not feel like I'm wasting my time. :P $\endgroup$ – Asaf Karagila Aug 19 at 17:04
  • $\begingroup$ @JyrkiLahtonen: I probably should clarify that I distinguish between the use of order-theoretic tools such as transfinite induction (which I think should not be considered "set theory") and the study of how to obtain such tools within ZFC (which should be considered "set theory"). So I am inclined to remove set-theory from threads in the former category. $\endgroup$ – user21820 Aug 20 at 3:34
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Let me preface this by saying that there is a thick borderline between and where things are not very clear cut, and at the end of the day require set theorists to actively clean up the tag choices.

(Because ultimately the tag is aimed to focus on what set theorists would consider set theory, rather than elementary set theory, and this does not have a very clean definition.)


Now, basic questions about the $\aleph,\beth$ and $\alpha,\beta$ of cardinals and ordinals (respectively) fall right into that space. And it's hard to give an exact answer when something is elementary and when it is not.

To me there are two aspects here:

  1. Is the question technical? Is the question about the inner working of the ordinals, or the cardinals? Is it just about some basic properties of their ordering or their arithmetic? The more technical stuff goes into the tag, and the more basic things go into . In either case, using the , , and even is probably a good idea.

  2. Is the answer requiring some nontrivial set theoretic insights? If the answer is yes, then by all means, this is a worthy question. If not, then no (assuming that the above point was also pointing towards "no"). Sometimes there is an elementary answer and a more complicated answer, and those are borderline cases and are harder to classify.

In summary

My answer here is absolutely useless to anyone who is not already a working set theorist, or at least a working logician, that developed a natural sense into what goes into what, especially for that gray area between the two tags. I do apologise for that, truly.

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  • $\begingroup$ I do have to wonder whether it even makes sense to have this distinction in the first place... But I'm not sure it would be a Good Idea to flood set-theory with basic "How do the De Morgan laws work again?" questions, so that leaves me feeling a bit stuck here. Perhaps we can simply draw an arbitrary line at the power set, and say that if you have more than two layers of nested sets before you get to opaque urelements, or if you don't have urelements at all, then it's set-theory? $\endgroup$ – Kevin Aug 22 at 3:26
  • $\begingroup$ We had that discussion on meta when the elementary tag was introduced. Since then I have grown to appreciate the separation a lot more. Unfortunately your suggestion is not a good one. $\endgroup$ – Asaf Karagila Aug 22 at 5:41
  • $\begingroup$ I agree that it is not a good suggestion, but I'm at a loss for what to suggest instead. "Only set theorists know which to use" is dreadfully unhelpful to new users who just want to ask a simple question. $\endgroup$ – Kevin Aug 22 at 5:44
  • $\begingroup$ Since we do not execute people for making mistakes, I don't see the problem. Moreover, the key here is the same as the MathOverflow vs. MSE. If you have to ask, it's almost probably elementary. But hey, why did we spend all that time writing tag wikis with guides as to what goes where... Am I right? $\endgroup$ – Asaf Karagila Aug 22 at 5:47
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When I learned set theory (in my final year of undergraduate study), we learned predicate logic first, then the axioms of ZFC as a worked example of predicate logic, and lastly constructed ordinals within ZFC and proved things about their basic properties. This involved some in-depth study of recursive definitions, $\in$-induction and so on, before we could even talk about the standard inductive definitions of ordinal arithmetic.

Now this might seem far removed from the way that many other people first learn about ordinals (including people who took the same course under different lecturers). The point is that any distinction between 'elementary set theory' and 'real set theory' is going to be arbitrary.

As far as ordinals go, I don't think they necessarily need to be in either tag. Countable ordinals, in particular, can be defined and dealt with inside any type theory that supports induction over natural numbers. Simple ordinal arithmetic can be treated as an axiomatic theory in its own right.

Once you start talking about uncountable ordinals and Hartogs's Lemma (and, by extension, the axiom of replacement), the choice of model becomes unavoidable. The question "What is the cardinality of $\omega_1$?" is perhaps an elementary one, but seems unavoidably linked to questions about forcing and independence. Given that there is a separate site for research-level questions, seems the more appropriate tag for such questions.

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  • $\begingroup$ Do you mean that basic questions about transfinite ordinals should not be tagged with either of the set-theory tags? $\endgroup$ – Henning Makholm Aug 22 at 10:42
  • $\begingroup$ @HenningMakholm It would depend on the question itself, but I don't see why a 'pure ordinal theory' question would need to be explicitly tagged with set-theory. It's not a problem if it is, since there are far more arbitrary groupings of topics into tags on this site. $\endgroup$ – John Gowers Aug 22 at 10:51
  • $\begingroup$ Did you learn with Oren or with Thomas? $\endgroup$ – Asaf Karagila Aug 22 at 10:52
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    $\begingroup$ Also, I think that what @Henning is asking in this thread, as a whole, is what would be a good guideline for deciding for each question whether or not it is [set-theory] or its elementary counterpart. Towards this, your answer is mostly relevant by saying "neither is also an option". $\endgroup$ – Asaf Karagila Aug 22 at 10:54
  • $\begingroup$ E.g., algebra-precalculus seems to be based on which subjects tend to be taught together, so I can understand ordinals coming under set-theory on the same grounds. $\endgroup$ – John Gowers Aug 22 at 10:54
  • $\begingroup$ @AsafKaragila Johnstone! $\endgroup$ – John Gowers Aug 22 at 10:54
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    $\begingroup$ My face got so blank for a moment there, I don't think I will ever be able to make another expression for the rest of the day. $\endgroup$ – Asaf Karagila Aug 22 at 10:55
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    $\begingroup$ "What is the cardinality of $\omega_1$?" might be a good touchstone -- except, alas, it is the expected answer rather than the question itself that is determinative. In elementary-set-theory a reasonable answer might be "It's either $|\mathbb R|$ or somewhere strictly between $|\mathbb N|$ and $|\mathbb R|$; we don't know". In set-theory it would be "why, that's $\aleph_1$, by definition; it cannot be anything else". $\endgroup$ – Henning Makholm Aug 22 at 11:04
  • $\begingroup$ Well, perhaps platonic-set-theory vs formal-set-theory makes it clearer. $\endgroup$ – John Gowers Aug 22 at 11:07
  • $\begingroup$ @JohnGowers: algebra-precalculus is a horrible name, kept for lack of a better alternative, but the meaning is quite clearly delimited from abstract-algebra -- one is about manipulation of single symbolic expressions; the other is about entire algebraic structures and the relations between them. $\endgroup$ – Henning Makholm Aug 22 at 11:08
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    $\begingroup$ @HenningMakholm I wasn't calling that into question, just pointing out that some of the topics within algebra-precalculus don't seem especially related to one another -- although certainly none of them could be called abstract algebra. $\endgroup$ – John Gowers Aug 22 at 11:12

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