Who was the first to mention that the well-ordering theorem implies the axiom of choice?

Question

It is well-known that Zermelo in 1904 used the axiom of choice (AC) to prove Cantor's well-ordering principle (WO). And one can find various surveys which list equivalences to AC. However, even with the help of Moore's nice book about AC I couldn't find out who first became aware of the fact that WO implies AC. I understand that this is trivial to prove and most likely not something you'd write a paper about, but it is still an interesting fact, isn't it? Was this really too obvious to mention even at that time, was nobody interested in such questions, or am I just missing that XY pointed this out in 19..?

Answer 1

See Origins and Chronology of the Axiom of Choice : Mathematical Applications of the Axiom of Choice :

The Axiom of Choice has numerous applications in mathematics, a number of which have proved to be formally equivalent to it. Historically the most important application was the first, namely:

The Well-Ordering Theorem (Zermelo 1904, 1908). Every set can be well-ordered.

After Zermelo published his 1904 proof of the well-ordering theorem from AC, it was quickly seen that the two are equivalent.

Some details from :

• Gregory Moore, Zermelo's Axiom of Choice : Its Origin Development and Influence, Springer (1982).

Page 93 : On 1 December 1904 Borel finished a brief article, requested by David Hilbert as an editor of Mathematische Annalen ["Quelques remarques sur les principes de la theorie des ensembles," MA 60, 194-195], on the question of Zermelo's [1904] proof. What Zermelo had done, Borel wrote, was to show the equivalence of the problems of:

(A) well-ordering an arbitrary set $M$, and

(B) choosing a distinguished element from each non-empty subset of $M$.

Page 116 : Schoenflies [in 1908] believed that Zermelo had shown only the equivalence of the Axiom of Choice and the proposition that every set can be well-ordered.

Page 161 : Russell added [in aletter of 1908] that in a forthcoming paper [1908] he had shown the Axiom of Choice to be equivalent to the Multiplicative Axiom.

Page 167 : Although over one hundred propositions have been proven equivalent to the Axiom of Choice, only two were publicly recognized as such by 1908: the Well-Ordering Theorem and the Multiplicative Axiom.

Page 170 : in 1915 [Hartogs] gived a new proof of the Well-Ordering Theorem. Hartogs established that, in Zermelo's system without the Axiom the Trichotomy of Cardinals implies that every set can be well-ordered. Since the 1890s mathematicians had recognized that if every set can be well-ordered, then Trichotomy holds. But the converse was not at all obvious to them. As late as 1908 Schoenflies had insisted that Trichotomy is essentially weaker than the Axiom. Similarly, before he learned of the Axiom, Jourdain had claimed that Trichotomy is weaker than the Well-Ordering Theorem [1904]. Yet three years later he considered Trichotomy to be equivalent to the Axiom and hence to the Well-Ordering Theorem as well. Nevertheless Jourdain gave no argument, not even a heuristic one, for this equivalence [1907]. It remained for Hartogs [1915] to provide a convincing proof.

Page 213 : When in 1918 Sierpihski proposed that mathematicians actively seek to determine the deductive strength of various propositions relative to the Axiom of Choice, only a few such propositions were known to be equivalent to it. Chief among these were the Well-Ordering Theorem, the Multiplicative Axiom, and the Trichotomy of Cardinals.

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Added Sept,8

It seems that the equivalence was already implicitly stated by Beppo Levi in 1904; see :

• Barbara Moss, Beppo Levi and the axiom of choice, Hist.Math, 1979.

Answer 2

It's difficult to prove a negative of this kind, but let me take a stab at arguing why we shouldn't expect to find such an explicit claim:

I think one important factor here is that by the time anyone would have asked that question, the situation was already better understood. My understanding is that "AC implies WO" was understood as proving a theorem from an axiom, that is, AC and WO were considered on fundamentally separate footing (and we can see this in the fact that attempts to justify AC on "logical" grounds weren't accompanied by attempts to similarly justify WO). The idea of looking the other way emerged, to my knowledge, with Tarski's result that AC is equivalent to $\vert X\vert^2=\vert X\vert$ for infinite $X$ (which famously had some publication issues); but at that point set theory was unquestionably developed to the point that "WO implies AC" would have been truly trivial. Even if it weren't obvious to Zermelo - and honestly I think it was, the danger of hindsight notwithstanding - I don't think there was a moment where it would have been simultaneously surprising and interesting enough to state.

Of course, I'm making some assumptions here, but I hope that the above gives additional credence to the view that you won't find what you're looking for.