Continuum hypothesis
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In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets. It states:
- There is no set whose cardinality is strictly between that of the integers and the real numbers.
Establishing the truth or falsehood of this hypothesis is the first of Hilbert's 23 problems presented in the year 1900. In 1963, Paul Cohen proved that the hypothesis is independent of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), a standard axiomatization of set theory, complementing earlier work by Kurt Gödel in 1940. Such independence means that ZFC can be augmented by either CH or its negation ¬CH, in both cases producing a system of axioms that is consistent if and only if ZFC is consistent.
The name of the hypothesis comes from the term the continuum for the real numbers. Cohen was awarded the Fields Medal in 1966 for his proof.
See also