How should basic questions about transfinite ordinals and cardinals be tagged?

Question

I was about to retag this question when I noticed that neither of the elementary-set-theory or set-theory 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 elementary-set-theory -- 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 set-theory. 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 set-theory 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 set-theory list, but it's also conceivable that we should really move them to elementary-set-theory instead, possibly together with basic questions about how to use the ZFC axioms.

Answer 1

Let me preface this by saying that there is a thick borderline between set-theory and elementary-set-theory 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 set-theory 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.)

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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 set-theory tag, and the more basic things go into elementary-set-theory. In either case, using the ordinals, cardinals, and even well-orders 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 set-theory 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.

Answer 2

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, set-theory seems the more appropriate tag for such questions.