What did Paul Cohen prove about whether the axiom of choice follows from Zermelo-Fraenkel set theory?

In 1963, the American mathematician Paul Cohen proved that the axiom of choice cannot be derived from the standard Zermelo-Fraenkel set theory (ZF) axioms.12

By demonstrating that the negation of the axiom of choice is consistent with the ZF axioms (assuming ZF itself is consistent), Cohen established that the axiom of choice is independent of ZF. This built upon earlier work by Kurt Gödel, who had previously shown that the axiom of choice is consistent with ZF. Together, these results confirm that the axiom of choice can neither be proved nor disproved using the other axioms of Zermelo-Fraenkel set theory.12

To achieve this, Cohen introduced a powerful new mathematical technique known as forcing, which has since become a fundamental tool in set theory for constructing models and proving the independence of various propositions.23