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Reading through the tag wikis for and suggests that the axiom of choice is improperly included in the former when it should be in the latter. Note that elementary-set-theory already includes cardinal arithmetic.

Clarification

I am not suggesting that applications of the axiom of choice are necessarily, or even usually, easy or suitable for high school students or beginning undergrads. But I think we can and should base these tags more on the difficulty of the concepts involved than on the difficulty of the proofs involved. An undergrad can read a difficult proof using the axiom of choice and understand each step, but may not grasp the structure of the whole. She will not, however, be likely to make head or tail of a forcing argument.

The tag excerpt for 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 tag wiki for elementary-set-theory:

This tag is for elementary questions on set theory—the sort of material covered in undergraduate set theory texts. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, countability and uncountability, power sets, transfinite induction, well-orders, ordinal and cardinal arithmetic, etc. More advanced topics should use the "set-theory" tag instead.

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    $\begingroup$ No.${}{}{}{}{}$ $\endgroup$ – Andrés E. Caicedo Dec 31 '13 at 5:44
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    $\begingroup$ @AndresCaicedo, that was a mightily unhelpful response. Can you give some reasoning as to why the axiom of choice fits better with things like forcing than things like cardinal arithmetic. $\endgroup$ – dfeuer Dec 31 '13 at 5:47
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    $\begingroup$ 1, 2, 3, 4, 5. $\endgroup$ – Andrés E. Caicedo Dec 31 '13 at 6:42
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    $\begingroup$ 6, 7, 8, 9, 10. $\endgroup$ – Andrés E. Caicedo Dec 31 '13 at 6:47
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    $\begingroup$ 11, 12, 13, 14, 15. $\endgroup$ – Andrés E. Caicedo Dec 31 '13 at 6:51
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    $\begingroup$ @Andres: Also 16, and 17. The former was developed into the main result of my M.Sc. thesis, and I don't like to think of it as particularly elementary. $\endgroup$ – Asaf Karagila Dec 31 '13 at 8:34
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I agree with Andres, along with the many examples posted in the comments.

The axiom of choice might be dealt with naively in some contexts, and those questions elicit the elementary tag. For example "How to use the axiom of choice to prove that every function has an injective inverse?"

But on the other hand, there are questions on the site related to the axiom of choice which are well suited to appear in research level papers, or theses and dissertations. For example questions related to symmetric extensions by forcing, or failure of certain set theoretical axioms in the absence of choice.

This is also the case with , by the way. Some of the questions fit the elementary tag, and others don't fit it very well. For this reason it's a separate tag.

Do note, however, that it is more common to find elementary questions about cardinal arithmetic, then it is to find elementary questions about the axiom of choice.

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  • $\begingroup$ I don't see how that can be, as the usual rules of cardinal arithmetic depend on the axiom of choice. $\endgroup$ – dfeuer Dec 31 '13 at 18:18
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    $\begingroup$ No, not really. The rules trivialize in the presence of choice, but they don't change. $\endgroup$ – Asaf Karagila Dec 31 '13 at 18:34

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