He is noted for inventing a technique called forcing which he used to show that neither the continuum hypothesis nor the axiom of choice can be proved from the standard ZermeloFraenkel axioms of set theory. In conjunction with the earlier work of Gödel, this showed that both these statements are independent of the ZermeloFraenkel axioms: they can be neither proved nor disproved from these axioms. For his efforts he won the Fields Medal.
This result is possibly the most famous nontrivial example illustrating Gödel's incompleteness theorem.
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