Or rather why do some people vehemently reject axiom of choice?

I am interested from this from the perspective of the philosophy of computation. Intuitively, from the little I know, it seems people are smuggling their intuitions about computation when they reject the Zorn's lemma. I would like a much clearer perspective on what each side says.

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    In the mathematical community, there is a wide (positive) consensus about AC. For an historical overview, see G.H.Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence. – Mauro ALLEGRANZA Jan 30 '18 at 10:04
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    Also useful: The Axiom of Choice. – Mauro ALLEGRANZA Jan 30 '18 at 10:13
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    Accepting Zorn's lemma equally smuggles in one's intuitions about finite sets. Most mathematicians "accept" it in the sense that it is handy, not that they find it intuitively self-evident or even remotely convincing. After Lebesgue non-measurable sets and the Banach-Tarski paradox almost no one feels that way. But only some are purists and have higher philosophical scruples, like intuitionists and constructivists. To them unintuitive even piecemeal (through construction) means out with it. – Conifold Jan 30 '18 at 23:23

One reason to reject AC is because one is a mathematical constructivist: Diaconescu showed that AC implies Excluded Middle (EM), and using either to produce results that merely assert "there must exist an X" rather than "here is how to construct an X" are not acceptable, from a constructive point of view.

Another reason people reject AC is that it can produce 'counter-intuitive' results such as the Banach-Tarski paradox or the Vitali set. But ¬AC can produce 'counter-intuitive' results as well, such as that the real numbers are a countable union of countable sets.

More broadly, there are other forms of choice available, such as Dependent Choice (DC), which can allow one to avoid some of the consequences of AC without losing 'too much' power to prove certain things.

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