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sjdfio

One does not need to invoke the axiom of choice to construct a choice function on a finite number of sets. This can be done using existential instantiation a finite number of times. Nor is the axiom necessarily needed to construct a choice function on an infinite collection of sets. For example, if the union of such a collection has a bijective correspondence with a well-ordered set. The axiom of choice is only needed when constructing a choice function would require infinitely many steps.
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Sorry about the typos! This is the first quote I submitted here, and was a stream of consciousness I typed out quickly. I didn't realize it would be difficult to remove/edit once I hit submit.

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