In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:...

The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that 2 ℵ 0 = 2 ℵ 1 {\displaystyle...

second smallest is ℵ 1 {\displaystyle \aleph _{1}} (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly...

argument may not be valid in some particular case. Consider the continuum hypothesis, stating that there is no set with size strictly between the size...

was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set...

believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable...

of the continuum hypothesis implies the Suslin hypothesis. The Suslin hypothesis is also independent of both the generalized continuum hypothesis (proved...

universe of set theory in which the continuum hypothesis must hold. In 1963, Paul Cohen showed that the continuum hypothesis cannot be proven from the axioms...

numbers. Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, assuming...

{\displaystyle \aleph _{0},\aleph _{1},\dots } ), but unless the generalized continuum hypothesis is true, there are numbers indexed by ℵ {\displaystyle \aleph } that...

in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity 2 ℵ 0 {\displaystyle \aleph _{0}}...

{\displaystyle (\aleph _{1},\aleph _{2},\aleph _{3},\ldots ).} His continuum hypothesis is the proposition that the cardinality c {\displaystyle {\mathfrak...

developed into -bl- in names like Ableca. The Western-Hispano Celtic continuum hypothesis received little support from linguists, who have widely rejected...

assume the generalized continuum hypothesis. If the continuum hypothesis holds, all real closed fields with cardinality of the continuum and having the η1...

of choice from the remaining Zermelo-Fraenkel axioms and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved...

{\displaystyle {\texttt {AX}}} is equivalent to the negation of the continuum hypothesis (CH). Sierpiński's theorem answered a question of Hugo Steinhaus...

for which the continuum hypothesis fails can be solved using statistical mechanics. To determine whether or not the continuum hypothesis applies, the Knudsen...

The term weak continuum hypothesis can be used to refer to the hypothesis that 2 ℵ 0 < 2 ℵ 1 {\displaystyle 2^{\aleph _{0}}<2^{\aleph _{1}}} , which is...

most notably, to the continuum hypothesis, which was proposed by mathematician Georg Cantor in 1873. The continuum hypothesis represents a tendency for...

set theory with the continuum hypothesis added as a further axiom, and the set theory with the negation of the continuum hypothesis added. Informally,...

paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an...

The continuum hypothesis is independent of ZFC, a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or...

extra axiom stating that there are no endpoints in the order. The continuum hypothesis is a statement in the language of ZFC that is not provable within...

the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis...

reasons and that would decide the continuum hypothesis. Many large cardinal axioms were studied, but the hypothesis always remained independent from them...

natural numbers. The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no...

statement that is independent of ZF. For example, the generalized continuum hypothesis (GCH) is not only independent of ZF, but also independent of ZFC...

multiverse views is the attitude to the continuum hypothesis. In the universe view the continuum hypothesis is a meaningful question that is either true...

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...

theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all...