Why did the axiom of choice cause controversy among mathematicians?

The controversy surrounding the axiom of choice primarily stems from its nonconstructive nature. Unlike other axioms in set theory, it asserts the existence of a set—or a "choice function"—without providing a specific algorithm or rule to construct or identify its elements.12

Key factors contributing to this debate include:

• Lack of Selection Rules: For infinite collections of sets where no natural selection rule exists (such as choosing one sock from each of infinitely many pairs), the axiom assumes a choice can be made, even though it cannot be explicitly defined.1
• Counterintuitive Results: The axiom leads to paradoxical conclusions, most notably the Banach-Tarski paradox, which suggests a solid sphere can be decomposed and reassembled into two spheres of the same radius.1
• Mathematical Philosophy: Historically, some mathematicians rejected the axiom entirely because it lacked a constructive proof. Others adopted a cautious approach, using it only when necessary.2

While some mathematicians still view its use as improper, the vast majority of the community accepts it as an indispensable tool due to its utility and consistency within the ZFC framework.12