In the "Equivalents" section, the abstract algebra equivalent seem completely natural to me. The set theory ones give it an air of being mystical and not being true, imo. However: "Every unital ring other than the trivial ring contains a maximal ideal." http://en.wikipedia.org/wiki/Axiom_of_choice
Stuff like that makes me so glad I veered off from mathematics to engineering.. a few calculus classes and never look back...
The AC is either extremely interesting to you, or a complete nussiance to be avoided or ignored. I am actually somewhere in between. It is often stated very simply as âThe Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.â â Bertrand Russell "The principle of set theory known as the Axiom of Choice has been hailed as âprobably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid's axiom of parallels which was introduced more than two thousand years agoâ (Fraenkel, Bar-Hillel & Levy 1973, §II.4)...." "...Zermelo's original purpose in introducing AC was to establish a central principle of Cantor's set theory, namely, that every set admits a well-ordering and so can also be assigned a cardinal number..." http://plato.stanford.edu/entries/axiom-choice/ AC is all about choice functions, and in particular there are very interesting applications to Godel, P=NP, Complexity Theory and Computer Science in General. In some sense, the question of the AC leads to strong and weak AI. This is probably the best book on the subject The Axiom of Choice (Dover Books on Mathematics) http://www.amazon.com/Axiom-Choice-Thomas-J-Jech/dp/0486466248/ref=pd_sim_b_3 There are still people that don't believe it: http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/
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