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...

uncountability proof). The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers...

, 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...

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

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

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...

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

{\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 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...

_{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...

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...

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...

complements, and taking the union of all that over all of ω1. The cardinality of the set of real numbers (cardinality of the continuum) is 2ℵ0. It cannot be...

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...

continuum hypothesis is the proposition that there are no intermediate cardinal numbers between ℵ 0 {\displaystyle \aleph _{0}} and the cardinality of...

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...

set of natural numbers. The cardinality of R is often called the cardinality of the continuum, and denoted by c {\displaystyle {\mathfrak {c}}} , or 2 ℵ...

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...

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

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)....

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

correspondence with the set of real numbers (see Cardinality of the continuum). The power set of a set S, together with the operations of union, intersection...

=2^{\aleph _{0}}.} In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of Lebesgue measurable sets...

mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values...

properties: The cardinality of any Cantor space is 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} , that is, the cardinality of the continuum. The product of two (or...

this yields a notional cardinality of the continuum, Hartman advises that when setting out to describe a person, a continuum of properties would be most...

\beth _{0}} : the cardinality of the continuum 2ℵ0 ℭ or c {\displaystyle {\mathfrak {c}}} : the cardinality of the continuum 2ℵ0 Omega: ω, the smallest infinite...

and another is the real numbers (isomorphic to any other set with the cardinality of the continuum). In fact, it has an infinite number of models, one for...

infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension. Many vector spaces that are considered in mathematics...