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I read that Zermelo formulated AC in 1904 in order to formally prove the well-ordering theorem.

Vitali’s 1905 proof of the existence of a non-measurable set of real numbers appears to the first “unpleasant” consequence of AC that is commonly cited. Q: Is this the first such example?

I also recall reading that someone has written a whole book on the unpleasant consequences of both AC and $\lnot$AC. Q: What is the first example obtained of an unpleasant consequence of $\lnot$AC, and what is the book whose name and author I cannot recall?

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  • $\begingroup$ Maybe Gregory Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence (1982, also Dover reprint). $\endgroup$ – Mauro ALLEGRANZA Oct 20 '16 at 18:49
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    $\begingroup$ According to Moore (page 185) "in 1914 appeared what seemed to be irrefutable evidence against the Axiom [of Choice] : Hausdorff's paradox." $\endgroup$ – Mauro ALLEGRANZA Oct 20 '16 at 18:57
  • $\begingroup$ @MauroALLEGRANZA Thanks for the reference. Moore's book looks interesting and it is not expensive ($28 at amazon.ca and probably cheaper via eBay). I may give it a go since it is a very interesting subject with an apparently interesting history. The book I recall was specifically dedicated to results deduced from AC and not(AC). I'm currently struggling to read Jech's text "The Axiom of Choice", but I currently lack a sufficient grounding in Set Theory to fully understand its contents. $\endgroup$ – Nick Oct 20 '16 at 19:20
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    $\begingroup$ Howard & Rubin, Consequences of the Axiom of Choice or Horst Herrlich, Axiom of Choice ? $\endgroup$ – Mauro ALLEGRANZA Oct 20 '16 at 19:33
  • $\begingroup$ @MauroALLEGRANZA Both of these texts are good candidates, especially Herrlich's, whose main sections are entitled "Disasters with Choice", "Disasters without Choice", and "Disasters either way". Thanks again for you help. It's tempting to give Herrlich a go, but I think that at my current level the historically based account of Moore may be more suitable. $\endgroup$ – Nick Oct 20 '16 at 20:23
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Vitali's construction is probably the first example "unpleasant" in the modern sense, but well-ordering of the continuum was unpleasant enough to some at the time of Zermelo's proof in 1904. Accepting that the continuum can be dissociated into points (contra Aristotle) was recent and hard enough, that it could be well-ordered strained credulity further. Borel and Lebesgue rejected it right away, and Poincare openly mocked the idea, Peirce and Weyl rejected even that the continuum is a set. Grothendieck topos would probably better fit what they had in mind than a set, but it is interesting that it was not proved until 1975 that AC implies the law of excluded middle, and hence is automatically out for intuitionists. To Peano the axiom of choice itself looked like an unpleasant principle, not to be admitted. Interestingly enough, Poincare accepted the axiom of choice itself but rejected Zermelo's proof on the grounds of "impredicativity" of the well-ordering, a self-referential aspect in its definition. The same issue earlier led Russell to develop the theory of types. Zermelo's response (to both) was that without impredicativity many classical constructions (like the least upper bound) fall apart as well.

The earliest compendium of axiom of choice zoo of pathologies is perhaps Sierpinski's long 1918 paper in French L'axiome de M. Zermelo et son róle dans la théorie des ensembles et l'analyse [Bulletin international de l'Académie des Sciences de Cracovie, Classe des sciences mathématiques et naturelles, Séie A, Sciences mathématiques, année 1918, pp. 97–152.] I do not think it was translated into English, but it started the enterprise of sifting through the classical mathematical theorems to determine whether or not they depend on AC, and looking for curiosities that follow from it or its negation (Sierpinski also did similar work for the continuum hypothesis). Polish mathematicians did a lot of work in these directions in 1918-1940, which is reflected in Sierpinski's Cardinal and Ordinal Numbers. In particular, the Banach-Tarski paradox came out of it, although it was a dramatization of the earlier Hausdorff's one from 1914. Another book not yet mentioned is Rubin-Rubin's Equivalents of the Axiom of Choice. Medvedev's Early History of the Axiom of Choice has a lot of early material with tracing of what does and does not depend on AC, but again it is not translated into English. Moore's book is perhaps the most accessible.

As for the negation of AC, Sierpinski pointed out in 1918 that the measurability of the countable union of measurable sets depends on (countable) AC. That was a nice complement to Vitali, nonmeasurable sets may be unpleasant but without any AC there will be no Lebesgue measure theory at all. But already Zermelo's proof showed that AC is equivalent to well-orderability of all sets, so without it cardinalities would be thrown into "chaos". That every cardinality is an alef (and in any case that continuum is) was Cantor's long-term conviction. He did not waver from it even when König at the 1904 International Congress of Mathematicians, in front of Cantor, presented his proof that the continuum could not be well-ordered. Cantor was sure that there must have been a mistake, then and there. Hausdorff and Zermelo soon realized that König made no mistakes, but there was a subtle gap in the proof of Bernstein's lemma which König used. So to Cantor not-AC would have been quite unpleasant. The eye of the beholder...

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  • $\begingroup$ Your answer had definitely inspired me to pursue the apparently rich and colourful history of AC and I think Moore's recent "more accessible" book will be a good place for me to start. May I ask a simple technical question? You use the word "construction" to describe Vitali's proof - so does Jech. Yet is it really a constructive proof. I ask this because Vitali applies AC to obtain a choice function on the equivalence classes, without explicitly stating a method of choice. Is it then simply a case of "given AC" we can "construct" a non-measurable set? Probably a dumb question! $\endgroup$ – Nick Oct 20 '16 at 23:26
  • $\begingroup$ Thinking about my previous comment, it does appear to be a dumb question. Obviously Vitali is proving non-measurability is a consequence of AC so he is going to need to invoke AC rather than bypassing it via an explicit choice function. $\endgroup$ – Nick Oct 20 '16 at 23:33
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    $\begingroup$ @NickR Not dumb at all.You are right, in ZFC the use of AC is almost the definition of non-constructive. Even pure existence proofs using excluded middle, which are also technically non-constructive, are intuitively nowhere near in ephemerality to anything touched by (uncountable) AC, like Vitali sets. Jech and others use "construction" in this case in mathematically colloquial rather than philosophical way: a series of steps that deliver existence of sought objects, even if some steps are simply postulated to work rather than "constructed", cf. ultrapower "construction" of hyperreals. $\endgroup$ – Conifold Oct 21 '16 at 0:09
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    $\begingroup$ Edited comment, to reflect the edit: Although Banach did not belong to the Warsaw school, but to the Lvov school, which had somewhat different interests, he did work on some problems in measure theory and referred to the ``Zermelo Axiom" and Hausdorff paradox even prior to his work with Tarski, e. g. here: Sur le probléme de la mesure, Fundamenta Mathematicae 4, 1923, s. 7-33, kielich.amu.edu.pl/Stefan_Banach/pdf/oeuvres1/08.pdf He credits Stanislaw Ruziewicz, a former student of Sierpinski and a professor in Lvov as his inspiration. $\endgroup$ – Margaret Friedland Oct 21 '16 at 1:47

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