Zermelo's Axiom of Choice
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…that consistently affirming or denying it has tripped up many world-class mathematicians! I am currently about 1/3 of the way through this really interesting history that tries to trace that development, Zermelo's Axiom of Choice by G.H. Moore (2013): http://www.amazon.com/gp/product/0486488411/ref=as_li_tl?ie=... (ToC: http://www.apronus.com/math/zermelos.htm)
on “The Axiom of Choice is Wrong (2007)” ↗…3 of the HoTT book). In this setting the axiom of choice says that for any set X, (∀ (x : X). ∥ P x ∥) → ∥ ∀ (x : X). P x ∥ (see section 3.8). Note that from the perspective of homotopy type theory, Zermelo's axiom of choice is too strong: it is equivalent to global choice (for all types A, ∥ A ∥ → A), which is inconsistent with univalence.
on “100 years of Zermelo's axiom of choice: What was the problem with it? (2006)” ↗