In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by (lowercase Fraktur "c") or .[1]
The real numbers are more numerous than the natural numbers . Moreover, has the same number of elements as the power set of Symbolically, if the cardinality of is denoted as , the cardinality of the continuum is
This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them.
Between any two real numbers a < b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (a,b) is equinumerous with This is also true for several other infinite sets, such as any n-dimensional Euclidean space (see space filling curve). That is,
The smallest infinite cardinal number is (aleph-null). The second smallest is (aleph-one). 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).
^"Transfinite number | mathematics". Encyclopedia Britannica. Retrieved 2020-08-12.
^Weisstein, Eric W. "Continuum". mathworld.wolfram.com. Retrieved 2020-08-12.
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theory, thecardinalityofthecontinuum is thecardinality or "size" ofthe set of real numbers R {\displaystyle \mathbb {R} } , sometimes called the continuum...
(see picture). By a similar argument, N has cardinality strictly less than thecardinalityofthe set R of all real numbers. For proofs, see Cantor's diagonal...
thecontinuum 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...
{\displaystyle \aleph _{0}} (thecardinalityofthe set of natural numbers), and thecardinalityofthecontinuum, that is, thecardinalityofthe set R {\displaystyle...
that the second beth number ℶ 1 {\displaystyle \beth _{1}} is equal to c {\displaystyle {\mathfrak {c}}} , thecardinalityofthecontinuum (the cardinality...
larger than the integers but smaller than the real numbers Cardinalityofthecontinuum, a cardinal number that represents the size ofthe set of real numbers...
complements, and taking the union of all that over all of ω1. Thecardinalityofthe set of real numbers (cardinalityofthecontinuum) is 2ℵ0. It cannot be...
_{0}}>{\aleph _{0}}} . Thecontinuum hypothesis states that there is no cardinal number between thecardinalityofthe reals and thecardinalityofthe natural numbers...
continuum hypothesis is the proposition that there are no intermediate cardinal numbers between ℵ 0 {\displaystyle \aleph _{0}} and thecardinality of...
field Κ of larger cardinality. Ϝ has thecardinalityofthecontinuum, which by hypothesis is ℵ 1 {\displaystyle \aleph _{1}} , Κ has cardinality ℵ 2 {\displaystyle...
because the number of choices for ⟨b2, b4, b6, ...⟩ has the same cardinality as thecontinuum, which is larger than thecardinalityofthe proper initial...
set of natural numbers. The cardinality of R is often called thecardinalityofthecontinuum, and denoted by c {\displaystyle {\mathfrak {c}}} , or 2 ℵ...
stated in the form that every uncountable set of reals has thecardinalityofthecontinuum. The Cantor–Bendixson theorem states that closed sets of a Polish...
the cardinalityofthecontinuum, whose value in ZFC may be any uncountable cardinalof uncountable cofinality (see Easton's theorem). Thecontinuum hypothesis...
=2^{\aleph _{0}}.} In fact, thecardinalityofthe collection of Borel sets is equal to that ofthecontinuum (compare to the number of Lebesgue measurable sets...
of cardinality κ {\displaystyle \kappa } . Then X {\displaystyle X} has cardinality at most 2 2 κ {\displaystyle 2^{2^{\kappa }}} and cardinality at most...
characterized up to isomorphism by its cardinality, and that any uncountable standard Borel space has thecardinalityofthecontinuum. Borel isomorphisms on standard...
correspondence with the set of real numbers (see Cardinalityofthecontinuum). The power set of a set S, together with the operations of union, intersection...
infinite-dimensional vector spaces, and many function spaces have thecardinalityofthecontinuum as a dimension. Many vector spaces that are considered in mathematics...
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