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...

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...

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...

correspondence with the set of natural numbers, i.e. uncountable sets that contain more elements than there are in the infinite set of natural numbers. While...

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

power set has 23 = 8 elements, as shown above. If S is infinite (whether countable or uncountable), then P(S) is uncountable. Moreover, the power set is...

{\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)...

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...

\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...

considered as subsets of the real numbers. The Cantor set is an example of an uncountable null set.[further explanation needed] Suppose A {\displaystyle...

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...

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

that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of ω...

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

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any...

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...

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...

cocountable topology on a countable set is the discrete topology. The cocountable topology on an uncountable set is hyperconnected, thus connected, locally...

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...

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...

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

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...

The set of all even integers, expressed in set-builder notation. In set theory and its applications to logic, mathematics, and computer science, set-builder...

the lower limit topology, no uncountable set is compact. In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous...

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

In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple...

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...

the reals form an uncountable set, of which the rationals are a countable subset, the complementary set of irrationals is uncountable. Under the usual...