Concept
Cardinality
Related concepts (28)
Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. The union of two sets A and B is the set of elements which are in A, in B, or in both A and B.
Dedekind-infinite set
In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.
Cantor's paradox
In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number. In informal terms, the paradox is that the collection of all possible "infinite sizes" is not only infinite, but so infinitely large that its own infinite size cannot be any of the infinite sizes in the collection.
Beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written , where is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers (), but unless the generalized continuum hypothesis is true, there are numbers indexed by that are not indexed by . Beth numbers are defined by transfinite recursion: where is an ordinal and is a limit ordinal.
Universal set
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 ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself.
Order type
In mathematics, especially in set theory, two ordered sets X and Y are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) such that both f and its inverse are monotonic (preserving orders of elements). In the special case when X is totally ordered, monotonicity of f already implies monotonicity of its inverse. One and the same set may be equipped with different orders.
Pigeonhole principle
In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there must be at least two right-handed gloves, or at least two left-handed gloves, because there are three objects, but only two categories of handedness to put them into. This seemingly obvious statement, a type of counting argument, can be used to demonstrate possibly unexpected results.
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a/b is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory. For example, the numbers 3 and 2 are commensurable because their ratio, 3/2, is a rational number. The numbers and are also commensurable because their ratio, , is a rational number.