mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...

elements than there are positive integers. Such sets are now called uncountable sets, and the size of infinite sets is treated by the theory of cardinal numbers...

that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which...

Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. Although the terms...

a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of...

are transcendental. A set is called uncountable if it is not countable. That is, it is infinite and strictly larger than the set of natural numbers. The...

existence theorem that there are such sets. Each Vitali set is uncountable, and there are uncountably many Vitali sets. The proof of their existence depends...

"countably infinite". Sets with cardinality strictly greater than ℵ 0 {\displaystyle \aleph _{0}} are called uncountable sets. Cantor's diagonal argument...

study of the Mandelbrot set remains a key topic in the field of complex dynamics. The Mandelbrot set is the uncountable set of values of c in the complex...

as subsets of the real numbers. The Cantor set is an example of an uncountable null set. It is uncountable because it contains all real numbers between...

contradiction that a countable model of first-order set theory could contain an uncountable set. The paradox arises from part of the Löwenheim–Skolem...

Cantor set a universal probability space in some ways. In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has...

{\displaystyle \operatorname {J} (f)} is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers)...

be the set of Gödel numbers of the true sentences about the constructible universe, with c i {\displaystyle c_{i}} interpreted as the uncountable cardinal...

terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction...

In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in...

In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the...

core model and satisfies the covering property, that is for every uncountable set x of ordinals, there is y such that y ⊃ x, y has the same cardinality...

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are...

\end{cases}}} The set of all such indicator functions, { 1 r } r ∈ R {\displaystyle \{\mathbf {1} _{r}\}_{r\in \mathbb {R} }} , is an uncountable set indexed by...

strong measure zero set has Lebesgue measure 0. The Cantor set is an example of an uncountable set of Lebesgue measure 0 which is not of strong measure zero...

the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories...

between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships...

sizes of two sets by setting them in one-to-one correspondence. His "revolutionary discovery" was that the set of all real numbers is uncountable, that is...

explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set. Another set F with the...

sets, αB will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω1, the first uncountable ordinal...

compact nor countably metacompact, hence not compact. Uncountable set: On any uncountable set, such as the real numbers R {\displaystyle \mathbb {R}...

infinite set of components is covered formally by allowing n = ∞ {\displaystyle n=\infty \!} . Where the set of component distributions is uncountable, the...

cofinite topology defined on an infinite set, as is the cocountable topology defined on an uncountable set. Pseudometric spaces typically are not Hausdorff...