How did the continuum hypothesis become independent of the usual axioms of set theory?

The continuum hypothesis (CH) became independent of the standard Zermelo-Fraenkel axioms (ZF) through the combined work of Kurt Gödel and Paul Cohen, which demonstrated that the hypothesis can neither be proved nor disproved within the system.123

In 1939, Gödel proved that CH is consistent with ZF by constructing a "constructive universe" of sets, showing that if ZF is consistent, then ZF plus CH is also consistent. This established that CH cannot be disproved.1234

In 1963, Cohen completed the proof of independence by introducing a technique called forcing. He demonstrated that, assuming ZF is consistent, ZF does not yield a proof of CH, meaning CH cannot be proved within the standard axioms.124

Because CH is neither provable nor refutable in ZF, it is considered independent of the usual axioms of set theory. This result has led to ongoing debates regarding whether new axioms should be adopted to settle the truth of the continuum hypothesis.243