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 that of the real numbers.
Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900. The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo–Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.
The name of the hypothesis comes from the term the continuum for the real numbers.
Read more about Continuum Hypothesis: Cardinality of Infinite Sets, Impossibility of Proof and Disproof in ZFC, Arguments For and Against CH, The Generalized Continuum Hypothesis
Famous quotes containing the words continuum and/or hypothesis:
“The further jazz moves away from the stark blue continuum and the collective realities of Afro-American and American life, the more it moves into academic concert-hall lifelessness, which can be replicated by any middle class showing off its music lessons.”
—Imamu Amiri Baraka (b. 1934)
“The hypothesis I wish to advance is that ... the language of morality is in ... grave disorder.... What we possess, if this is true, are the fragments of a conceptual scheme, parts of which now lack those contexts from which their significance derived. We possess indeed simulacra of morality, we continue to use many of the key expressions. But we have—very largely if not entirely—lost our comprehension, both theoretical and practical, of morality.”
—Alasdair Chalmers MacIntyre (b. 1929)