theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers R {\displaystyle \mathbb {R} } , sometimes called the continuum...

the cardinality of a set is the number of its elements. The cardinality of a set may also be called its size, when no confusion with other notions of...

{\displaystyle \aleph _{0}} (the cardinality of the set of natural numbers), and the cardinality of the continuum, that is, the cardinality of the set R {\displaystyle...

the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose cardinality is...

, the cardinality of the power set of the natural numbers. The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis...

cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced...

_{0}}>{\aleph _{0}}} . The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers...

field Κ of larger cardinality. Ϝ has the cardinality of the continuum, which by hypothesis is ℵ 1 {\displaystyle \aleph _{1}} , Κ has cardinality ℵ 2 {\displaystyle...

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...

that the second beth number ℶ 1 {\displaystyle \beth _{1}} is equal to c {\displaystyle {\mathfrak {c}}} , the cardinality of the continuum (the cardinality...

because the number of choices for ⟨b2, b4, b6, ...⟩ has the same cardinality as the continuum, which is larger than the cardinality of the proper initial...

integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for Georg Cantor...

larger than the integers but smaller than the real numbers Cardinality of the continuum, a cardinal number that represents the size of the set of real numbers...

the cardinality of the continuum, whose value in ZFC may be any uncountable cardinal of uncountable cofinality (see Easton's theorem). The continuum hypothesis...

with the cardinality of the continuum, and in his 1901 inaugural dissertation Bernstein proved that such a family can have no higher cardinality. Plotkin...

stated in the form that every uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish...

taking the union of all that over all of ω 1 . {\displaystyle \omega _{1}.} The cardinality of the set of real numbers (cardinality of the continuum) is...

as the infinite set of its points, and it is commonly taught that there are more real numbers than rational numbers (see cardinality of the continuum)....

all cardinals less than the cardinality of the continuum, 𝔠, behave roughly like ℵ0. The intuition behind this can be understood by studying the proof...

cardinality of the continuum equals the cardinality of the power set of the natural numbers, that is, the set of all subsets of the natural numbers. The statement...

infinite cardinal numbers, ℵ0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum). The theory was...

of the continuum (the cardinality of the set of real numbers): or equivalently that ℵ 1 {\displaystyle \aleph _{1}} is the cardinality of the set of real...

of cardinality κ {\displaystyle \kappa } . Then X {\displaystyle X} has cardinality at most 2 2 κ {\displaystyle 2^{2^{\kappa }}} and cardinality at most...

not hold for the cardinality of the continuum. The Erdős–Dushnik–Miller theorem has been called the first "unbalanced" generalization of Ramsey's theorem...

subset of the real numbers R {\displaystyle \mathbb {R} } . However, it has the same size as the whole set: the cardinality of the continuum. Since the real...

characterized up to isomorphism by its cardinality, and that any uncountable standard Borel space has the cardinality of the continuum. Borel isomorphisms on standard...

the requirement of continuity, then all smooth manifolds of bounded dimension have equal cardinality, the cardinality of the continuum. Therefore, there...

}\mid \alpha <\beta \rangle } , where β is an ordinal with the cardinality of the continuum. Let v0 equal r0. Then let v1 equal rα1, where α1 is least...

strictly larger than the cardinality of the continuum (i.e., set of all real numbers). This fact is easily verified by cardinal arithmetic: c a r d (...

cardinality of ⁠ R {\displaystyle \mathbb {R} } ⁠ is often called the cardinality of the continuum, and denoted by c {\displaystyle {\mathfrak {c}}} , or 2 ℵ 0...