# What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

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?

• Maybe Gregory Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence (1982, also Dover reprint). – Mauro ALLEGRANZA Oct 20 '16 at 18:49
• According to Moore (page 185) "in 1914 appeared what seemed to be irrefutable evidence against the Axiom [of Choice] : Hausdorff's paradox." – Mauro ALLEGRANZA Oct 20 '16 at 18:57
• @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. – Nick R Oct 20 '16 at 19:20
• Howard & Rubin, Consequences of the Axiom of Choice or Horst Herrlich, Axiom of Choice ? – Mauro ALLEGRANZA Oct 20 '16 at 19:33
• @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. – Nick R Oct 20 '16 at 20:23

• 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. – Margaret Friedland Oct 21 '16 at 1:47