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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 S. E..

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    $\begingroup$ Every lecturer in my first year courses explicitly stated that they were using the axiom of choice, so I think it depends on the lecturer/university. $\endgroup$
    – Huy
    Dec 16 '14 at 17:40
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    $\begingroup$ Speaking only for myself, I think it's not especially productive to hung-up with AC in "ordinary mathematics", especially if you're not going to distingush between countable choice and arbitrary choice (and there are many more subtler versions as well -- see near the end of my comments here) and especially if you don't have much background in mathematical logic (I don't, by the way) where one understands the distinction between non-constructively choosing an element from a nonempty set vs. AC itself. $\endgroup$
    – Dave L Renfro
    Dec 16 '14 at 19:18
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    $\begingroup$ Did your teachers explicitly point out all the other set-theoretic axioms they used? In many courses, one doesn't point out the use of obvious set-theoretic facts, and, for most people, the axiom of choice is quite obvious. Indeed, it had been used in lots of proofs before Zermelo pointed out that, to be rigorous, one needed to have it as an axiom. $\endgroup$
    – Andreas Blass
    Dec 16 '14 at 20:41
  • $\begingroup$ Depending on one's philosophical approach or sense of aesthetics, the full axiom of choice may not even be a desirable part of the foundations of mathematics. AC has a completely different character from some of your other examples because people think of it as part of one approach to building the universal foundation of all of mathematics. The Archimedean property is not foundational; there are systems besides the reals that satisfy it, and ever since Newton and Leibniz people have been using number systems that don't satisfy it. $\endgroup$
    – user507
    Dec 16 '14 at 21:01
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    $\begingroup$ As a mathematician whose research is in finite geometry, the axiom of choice is completely irrelevant to me. Also, when we first introduce vector spaces, they are usually all $\mathbb{R}^n$ or subspaces thereof so you don't need the axiom of choice for basis. $\endgroup$
    – DavidButlerUofA
    Dec 17 '14 at 6:21
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It is at once a philosophical, mathematical, foundational, and pedagogical issue.

Rather than expand on the first sentence, I recommend Gregory Moore's book which discusses some of the first three parts (and perhaps the fourth also). As a practical matter, the Axiom of Choice is a challenge to formulate precisely and formally in a first-order language, where as most of the other axioms are easy to state in such a language. Also, in practice much mathematics outside mathematical logic and some parts of computer science only needs to acknowledge the use (if any) of the axiom; the application usually does not concern foundational issues such as weak systems of arithmetic or set theory.

If you are trying to highlight or raise the awareness level of foundational issues, make note of the Axiom of Choice and suggest a reference or two for those who are interested. Spending more time on it in the context of almost any mathematics class is time lost to spending on the class material proper.

Gerhard "Pedagogical Practice Has Few Grounds" Paseman, 2014.12.16

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    $\begingroup$ What is the name of Gregory Moore's book? $\endgroup$
    – Bysshed
    Dec 17 '14 at 2:11
  • $\begingroup$ The title starts with "Zermelo's Axiom of Choice: Its Origins, Development,..." and has some other words in it. I think it is part of a Springer Series on History of Mathematics that was published in the 1980's. I figured a web search using the author's name followed by axiom of choice would work. Gerhard "Ask Me About WebSearch Compression" Paseman, 2014.12.16 $\endgroup$
    – Gerhard Paseman
    Dec 17 '14 at 4:05
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    $\begingroup$ @DanielKelsall G.H. Moore writes some good stuff about the history of mathematics. His AC Book is nice; I also like the first two (about CH) listed on MO here. $\endgroup$
    – Benjamin Dickman
    Dec 17 '14 at 4:28
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Much of mathematics works perfectly well with so-called "naive" set theory. That is, working with sets without worrying too much exactly what sets are. This isn't philosophically that different from calculus courses where you work with real numbers without worrying so much what they are (equivalence classes of Cauchy sequences of rational numbers), or for a more advanced example, modern homotopy theory research which uses $(\infty, n)$-categories without being too bothered by exactly what these are. These are all things that you generally have some intuition for, and until that intuition fails, it's okay to rely on it at first pass.

At some point, students (and mathematicians) need to understand the basics on which the theory is founded, but that isn't necessarily the beginning of their studies. Mathematics isn't just about going down to the foundations and checking that everything is rigorous; you also want to prove theorems once in a while, and sometimes hiding the foundational difficulties until later lets you get things done faster and more intuitively. This is no different from the fact that you weren't taught the Peano axioms of arithmetic in primary school, even though you were learning about the natural numbers. Sure, it wasn't completely rigorous, but finger-counting made it a whole lot easier to prove that $3+4=7$.

The right time to discuss the axiom of choice in any depth seems to me to be at the same point that you begin discussing the other axioms of ZFC and set up an axiomatic foundation of set theory. There's nothing really special about choice; it's an axiom that almost every mathematician is fine with accepting or to whom it doesn't even matter. If you try to talk about the axiom of choice without any discussion of the ZF axioms, you're still using naive set theory, and you aren't avoiding any of the paradoxes that axiomatic set theory was made to circumvent. Giving special privilege to AC isn't any more rigorous than not discussing it at all, just more indecisive.

That isn't to say that mentioning its use in passing does much harm; it may even be overall beneficial to mention such things. The majority of students won't care either way, but there may be a minority who are interested in other foundations besides ZFC, and that minority might be quite interested in the fact that every model of ZF¬C has vector spaces without bases. But spending significant lecture time on it in a course on e.g. linear algebra isn't a good use of the time. Interested students will likely study it on their own (now or later), and the rest of the class doesn't particularly need to know that the existence of bases for arbitrary vector spaces requires choice to prove the spectral theorem or put a matrix in Jordan normal form.

For me, this was in first-year undergraduate courses (and in some cases even in high school). For some other people I know, they didn't learn axiomatic set theory until graduate school. But for what it's worth, those people are now working mathematicians, while I'm a physicist; you can draw your own conclusions on that.

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