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The continuum hypothesis was advanced by Georg Cantor in 1878, [1] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being ...
the continuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC. The following four independence results are also due to Gödel/Cohen.);
The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers , ℵ 0 {\displaystyle \aleph _{0}} , or alternatively, that c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} .
Linear continuum, any ordered set that shares certain properties of the real line; Continuum (topology), a nonempty compact connected metric space (sometimes a Hausdorff space) Continuum hypothesis, a conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets and the cardinality of the set of all real ...
The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between and , means that =. [2] The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).
Suslin hypothesis; Remarks: The consistency of V=L is provable by inner models but not forcing: every model of ZF can be trimmed to become a model of ZFC + V=L. 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.
In this sense, the continuum hypothesis is undecidable, and it is the most widely known example of a natural statement that is independent from the standard ZF axioms of set theory. For his result on the continuum hypothesis, Cohen won the Fields Medal in mathematics in 1966, and also the National Medal of Science in 1967. [12]
The continuum hypothesis is equivalent to =. The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., = for all ordinals .