Formal Sciences & Matter / Mathematics

Continuum Hypothesis Is Undecidable

Paul Cohen's forcing method (1963) shows that the continuum hypothesis cannot be proved from the standard axioms of set theory, completing Gödel's result that it cannot be disproved.

Cantor asked in 1878 whether any set lies in size between the whole numbers and the real numbers, and Hilbert put the question first on his list of 1900. Gödel showed in 1938-40 that the standard axioms (ZFC) cannot refute it; in 1963 Paul Cohen invented forcing, a way of building new models of set theory, and proved they cannot prove it either, which earned him the 1966 Fields Medal. Forcing became the central tool of set theory; some mathematicians look for new axioms that settle the question, others hold that it has no single answer.