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

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

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...

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

in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity 2 ℵ 0 = ℵ 1 {\displaystyle 2^{\aleph...

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...

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

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...

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...

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

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

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

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...

aleph, aleph-one ℵ 1 {\displaystyle \aleph _{1}} , is known as the continuum hypothesis, which has been shown to be unprovable in standard set theories such...

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

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

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

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...

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...

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

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

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...

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...

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

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

axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analytical (in fact,...

set theoretic statements are independent of ZFC, among others: the continuum hypothesis or CH (Gödel produced a model of ZFC in which CH is true, showing...

even if one assumes the continuum hypothesis. In fact, it remains undecidable even under the generalised continuum hypothesis. The Whitehead conjecture...