The tag wikis say:
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.
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.