Why is the axiom of choice unnecessary for a finite collection of sets?

The axiom of choice is unnecessary for a finite collection of sets because the process of selecting an element from each set is guaranteed to conclude in a finite number of steps.1

In set theory, the axiom of choice is required only when dealing with infinite collections where no explicit algorithm or selection rule exists to identify a specific element from every set. For finite collections, one can simply choose elements one by one until the process is complete, as there is no risk of the task continuing indefinitely.1

As noted by Bertrand Russell, while choosing one sock from each of infinitely many pairs requires the axiom of choice due to the lack of a distinguishing rule, choosing one shoe from each pair does not, because a rule (such as always picking the left shoe) can be applied to each finite or infinite set of shoes. Thus, for any finite collection, the ability to perform a finite sequence of choices makes the axiom redundant.1