Continuum Hypothesis

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 HypothesisCardinality of Infinite Sets, Impossibility of Proof and Disproof in ZFC, Arguments For and Against CH, The Generalized Continuum Hypothesis

Other articles related to "continuum hypothesis, continuum, hypothesis":

The Generalized Continuum Hypothesis - Implications of GCH For Cardinal Exponentiation
... Although the Generalized Continuum Hypothesis refers directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation in all cases ...
Hyperreal Number - Development - The Ultrapower Construction
... field contains R it has cardinality at least that of the continuum ... This question turns out to be equivalent to the continuum hypothesis in ZFC with the continuum hypothesis we can prove this field is unique up to order isomorphism, and in ZFC with the negation of continuum ...
Cardinality Of The Continuum - The Continuum Hypothesis
... The famous continuum hypothesis asserts that is also the second aleph number ... In other words, the continuum hypothesis states that there is no set whose cardinality lies strictly between and This statement is now known to be independent of the axioms of Zermelo–Fraenkel set theory with ... That is, both the hypothesis and its negation are consistent with these axioms ...
Philosophical Problems - Philosophy of Mathematics - Mathematical Objects
... considering specific examples, such as the "continuum hypothesis" ... The continuum hypothesis has been proven independent of the ZF axioms of set theory, so according to that system, the proposition can neither be proven true nor proven false ... A formalist would therefore say that the continuum hypothesis is neither true nor false, unless you further refine the context of the question ...
Zermelo–Fraenkel Set Theory - Metamathematics - Independence in ZFC
... statements are independent of ZFC Continuum hypothesis Diamond principle Suslin hypothesis Martin's axiom (which is not a ZFC axiom) Axiom of Constructibility (V=L) (which is also not ... The Diamond Principle implies the Continuum Hypothesis and the negation of the Suslin Hypothesis ... Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis ...

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