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I've recently discovered that the following theorems require the axiom of choice to be proven:

  • every surjective function has a right inverse.
  • a real-valued function that is sequentially continuous at a point is necessarily continuous in the neighbourhood sense at that point.
  • every vector space has a basis.

When I revisited the proofs I was taught in first year, I was surprised that my lecturers had used the axiom of choice without declaring so.

It seems strange that so much effort was dedicated to establishing that mathematics is a rigorous subject [indeed much time was spent on learning the field axioms, well-ordering axioms, Archimedean principle and (later) the completeness principle] but to ignore the axiom of choice.

I am interested if there are reasons for omitting to mention the axiom of choice. Are there pedagogical reasons? Is it deemed too complicated? Is it more contentious than the other axioms?

Question also asked at Mathematics Educators S.E.

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    $\begingroup$ I think this question can and should be generalized to "Why isn't mathematics taught starting from the foundations?" because there's nothing particularly special about the AC. It's also probably a better suit for Mathematics Educators S.E.. $\endgroup$ – Git Gud Dec 16 '14 at 17:16
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    $\begingroup$ At my university it was never mentioned in the first 4 semesters. But I do not see this as a problem. We never really talked about ZF(C) in the first 4 semesters. After all, why do you care about the axiom of choice, but not about the power set axiom, the "selection" axiom (i.e. $\{x \in M \mid x\text{ has some property}\}$ is a set), ...? What you do/we did is to use naive set theory, but we were rigourous about the rest, so about the axioms of a group/vector space/complete ordered field, ... I think the axioms of set theory would only distract from the matter at hand. $\endgroup$ – PhoemueX Dec 16 '14 at 17:27
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    $\begingroup$ @Daniel: Most mathematics is not actually done on a strict axiomatic basis. In particular, formal axiomatizations of set theory are beside the point for most mathematics (and I speak as one for whom they are not beside the point). $\endgroup$ – Brian M. Scott Dec 16 '14 at 23:31
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    $\begingroup$ @DanielKelsall, no, it is just the acceptance of the fact that when you are struggling to understand the difference between uniform continuity and pointwise continuity for example (and this is a very, very advanced example, in fact), you have much better things to worry than axiomatic set theory. I can't think of anything more absurd than explicitly pointing out uses of the axiom of choice to people learning one-variable analysis, really... $\endgroup$ – Mariano Suárez-Álvarez Dec 16 '14 at 23:37
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    $\begingroup$ @Mariano: I disagree with that, on a fundamental level. Sure, if you're teaching engineering students or physics majors then there's usually no need to say "Hey, there's actually a deep thing going on here". But for math majors it's important to point out certain things. The equivalence of Zorn's lemma and the axiom of choice, or the well-ordering theorem, those are deep and fundamental theorems about the lack of constructivity when it comes to modern mathematics. If you treat it as an anecdote, the students will learn to treat set theory as an anecdote. [...] $\endgroup$ – Asaf Karagila Dec 17 '14 at 21:25
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I talked about this literally today with my students, since we finally arrived to the discussion where the axiom of choice is openly assumed.

First of all, in courses which are not set theory, the axiom of choice is usually chucked aside. Who didn't construct a sequence by recursion in calculus? Or proved that $f$ is continuous at $x$ if and only every sequence $x_n\to x$ satisfies $f(x_n)\to f(x)$?

Those things, in their generality require choice. If you get to courses about logic, the compactness theorem needs and uses choice, and in algebraic structure the proof that every unital ring has a maximal ideal requires choice as well. Many other proofs use choice without any consequence. And nothing bad happens.

But in set theory the axiom of choice has a special place, from two accounts:

  1. Historically the axiom of choice was controversial. The reason is that its implications are vast and many of them are counterintuitive. Of course, the negation of the axiom of choice is also strange and counterintuitive, how can you have non-empty sets without being able to choose from them at once?

    The reason, of course, is that it is not the axiom of choice or its negation which are counterintuitive. It is infinite sets, and infinite objects which baffles our finitary minds.

  2. Pedagogically, until the axiom of choice enters the discussion openly, you usually avoid explicit axioms, or that you do your best to allow some naiveté in the proofs (even when the axioms are mentioned). But everything that you prove to exist comes from specific formula.

    You want to show that $\Bbb Q$ is countable? Find a surjection from $\Bbb N$ onto $\Bbb Q$, and show that this suffices to establish that result.

    You want to show that $\Bbb R$ and $\mathcal P(\Bbb N)$ are equipotent? Write injections for both sides, and prove the Cantor-Bernstein theorem which essentially gives you a recipe for a bijection, given the two injections.

    But with the axiom of choice it becomes different. You are taking for granted some choice function exists. How does it work? What does it do? Nobody knows. Not only that, different choice functions will often give out different result in many constructions, which is fine, but the non-canonicity is deep-seated in the proofs here.

    And that is both an important thing to point out to students, as well an issue for confusion when you expect the student to be able and identify what sort of arguments were used at each point and turn.

So the axiom of choice can be easily assumed, and in fact in many places it is assumed, even implicitly, from the start. But if you want to properly train people to understand set theoretic subtleties then separating the axiom of choice is usually a good thing.

As a result, in some universities, and in some courses, the use of the axiom of choice is minimized when it is not totally necessary.

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    $\begingroup$ To your students of what? $\endgroup$ – Mariano Suárez-Álvarez Dec 16 '14 at 23:42
  • $\begingroup$ Of the naive set theory course I am teaching (well, giving the exercise classes. But I teach them new stuff there anyway). $\endgroup$ – Asaf Karagila Dec 17 '14 at 5:36
  • $\begingroup$ Notice that emphasizing this in a course on set theory, naive or not, is of course rather reasonable :-) $\endgroup$ – Mariano Suárez-Álvarez Dec 17 '14 at 16:48
  • $\begingroup$ Mariano, and I did say that in non-set theory courses these issues of choice are often ignored. The minor things (like dependent choice) will usually be assumed implicitly, and sometimes they are falsely claimed not to be needed, but stronger things like Zorn's lemma needed to show all sort of things might be casually given as a fact of mathematics. The rest of the answer deals with the set theory course. Especially the naive one, since the axiomatic set theory courses are usually constructed to deal with axioms specifically. $\endgroup$ – Asaf Karagila Dec 17 '14 at 17:58

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