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Is there some systematic study of which mathematical results requires the axiom of choice? A book or a review article. I don't mean only results without which some areas of mathematics would be hard to work with (for example, that $\mathbb{R}$ is Fréchet whose demonstration relies on the axiom of choice if I am not mistaken). But also the various kind of "monster" made possible by the axiom of choice (for example, non-Lebesgue-measurable subsets of $\mathbb{R}$).

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The book "Consequences of the axiom of choice" is excellent, but rather high level. Perhaps more accessible is Herrlich's wonderful book, which was what I learned from initially. Jech also has a book on the axiom of choice, but this focuses more on building models where choice fails/holds in specific ways, not general applications/implications of the axiom.

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  • $\begingroup$ The chapter titles are enticing indeed: Disasters without choice, Disasters with choice, Disasters either way. $\endgroup$ – user450847 Jun 5 '17 at 19:28
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There's the book 'Consequences of the Axiom of Choice' which can be found here.

Maybe there is some pdf online or something that's cheaper or available for free, but this definitely fits the bill.

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