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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.
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.
