Britannica Sources used in this answer

Why did mathematicians develop Zermelo-Fraenkel set theory with the axiom of choice?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Infinity — infinity Mathematical symbol representing the concept of infinity.

Mathematicians developed Zermelo-Fraenkel set theory with the axiom of choice (ZFC) primarily to provide a rigorous foundation for set theory, particularly when dealing with infinite sets.

The axiom of choice was explicitly formulated by Ernst Zermelo in 1904 to prove the well-ordering theorem, which states that every set can be well-ordered. A well-ordering is significant because it allows mathematicians to use transfinite induction, a process analogous to mathematical induction, to prove properties for elements of arbitrary sets.

While the axiom is not required for finite sets, it is essential for infinite collections where no specific algorithm or selection rule exists to pick elements. It asserts the existence of a "choice set" or "choice function" without requiring a constructive method to define it. Despite early controversy regarding its nonconstructive nature, ZFC has become the standard framework for modern mathematics because of its utility in developing theories of real numbers and transfinite cardinal numbers.