# When do we include "Axiom of Choice" as a tag?

My question is the following:

Suppose that the question is about a functional equation, where assuming the function we seek is continuous, we can show the existence of one particular type solution, whereas if continuity is not assumed, then the fact that $\mathbb R$ possesses a Hamel basis (as a linear space over $\mathbb Q$), guarantees the existence of discontinuous solutions of the functional equation.

And my question is the following: It is non-appropriate to include the "Axiom of Choice" among the tags?

It is important to say that many, if not most, of the subscribers who try to solve such functional equations are not familiar the Axiom of Choice/Zorn's Lemma, as such problems are popular in the IMOs and similar competitions.

• – Asaf Karagila Mod
Jan 25 '14 at 20:30

I think the prevailing view is that since most mathematicians consider the Axiom of Choice a "fundamental truth" (or at least use it without much consideration), the tag should only in included when Choice is somehow of more central interest to the question itself.

Examples of such question templates would be:

• How do you prove that $\Phi$ is equivalent to the Axiom of Choice?
• Can $\Psi$ be proven without the Axiom of Choice?
• How much Choice is needed to prove $\Xi$?
• Does $\Theta$ imply Choice?

Just as I would not include the tag for most questions dealing with cardinal arithmetic, even in cases where $\mathsf{AC}$ and $\neg \mathsf{AC}$ may yield different answers (such as this recent example), I would not include it in questions from other mathematical areas where whether or not Choice holds could possibly change the answer (and the OP has not indicated interest in the connection between Choice and the particular problem).

Of course, this doesn't preclude users from submitting answers which point out the connection between Choice and the given question. Such answers may be quite enlightening (though admittedly possibly not to the OP).

• How about: "This functional equation has a discontinuous solution, but this requires Zorn's Lemma" - What happens then? Jan 25 '14 at 21:03
• @Yiorgos: How about you just post the link to the question?
– Asaf Karagila Mod
Jan 25 '14 at 21:10
• See for example: math.stackexchange.com/questions/641416/… Jan 25 '14 at 21:13
• @Yiorgos: If your question was "Do discontinuous such functionals exist without assuming the Axiom of Choice," then, yes, I would include the axiom-of-choice tag. But I would rephrase your question to explicitly state that this is your central interest. As it stands, your Update seems to say that Choice yields a solution to your question, but this does not make your question about Choice. Jan 25 '14 at 21:23
• Remark: much of the advice of this answer is independent of the axiom-of-choice tag. The phrase "when [blank] is somehow of more central interest to the question itself" is the general guideline for tagging. Another example: don't tag a linear PDE question with semigroups just because solutions can be expressed in terms of a continuous semigroup of mappings; use it when somehow the structure of a semigroup is critical to the question. Jan 27 '14 at 8:35

I think an "axiom of choice equivalents" might be a reasonable tag. This includes gobs of important stuff.

• I'm against this. I wrote on this in the thread linked on the comments of the question.
– Asaf Karagila Mod
Jan 27 '14 at 1:16