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How did Paul Cohen contribute to the study of the continuum hypothesis?

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Infinity — infinity Mathematical symbol representing the concept of infinity.

In 1963, American mathematician Paul Cohen made a landmark contribution to set theory by proving that the continuum hypothesis is independent of the standard Zermelo-Fraenkel axioms (ZF).

While Kurt Gödel had previously shown in 1939 that the continuum hypothesis could not be disproved within the ZF system, Cohen demonstrated that it also could not be proven. By establishing that the hypothesis is neither provable nor disprovable under these axioms, he confirmed its independence.123

To achieve this, Cohen developed a revolutionary mathematical technique known as forcing. This method allows for the construction of models of set theory and has since become a fundamental tool for proving the independence of various other propositions in the field. Cohen’s work, for which he was awarded the Fields Medal in 1966, fundamentally shifted the understanding of axiomatic set theory, leading many mathematicians to conclude that the current axioms do not provide a complete description of mathematical reality.13