Set theory

branch of mathematics that studies sets, which are collections of objects

well-defined mathematical collection of distinct objects

operation denoted by symbol “∪” applied on two sets; the set of all distinct elements in the collection

combination of a set and an extra structure on it; more precisely, an object of a concrete category

in set theory, the theorem that a set has a strictly smaller cardinality than its powerset

proof technique in set theory

mathematical collection of sets that can be defined based on a property of its members (set theory)

the smallest superset of a given set that is closed under a given operation

one of several theories of sets used in the discussion of the foundations of mathematics; defined informally, in natural language

mathematical concept

partial order where all elements can be compared

part of the domain of a mathematical function

in set theory, a set large enough such that most ordinary mathematical constructions can take place within it

cardinality of the set of real numbers

subset of a measure space that is contained in a measurable set of measure zero

in set theory, a set whose elements are all subsets

set whose elements are the real numbers

undefined term motivated informally, usually by an appeal to intuition and everyday experience, or introduced axiomatically and eventually generated only by a series of elementary operations

In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. Formally, \operatorname{cf}(A) = \inf \{|B| : B \subseteq A, (\forall x \in A) (\exists y \in B) (x \leq y)\}

number of times an element appears in the multiset

theorem

set of arguments where two or more functions have the same value

the set of objects to which a term or concept applies

axiom in mathematical logic that all cardinals less than the cardinality of the continuum behave like ℵ₀ in a specific sense

In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same. ==In mathematics== The extensional definition of function equality, discussed above, is commonly used in mathematics. A similar extensional definition is usually employed for relations: two relations are said to be equal if they have the same extensions.

subset of a preordered set that includes all successors of its elements

combinatorial principle that there exists a family of sets 𝐴(𝛼)⊆𝛼 for 𝛼<ω₁ such that for any 𝐴⊆ω₁, the set of 𝛼’s with 𝐴∩𝛼=𝐴(𝛼) is stationary in ω₁

describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation

function which encodes two natural numbers into a single natural number

given a set, the least ordinal such that there is no injection from the ordinal into the set

mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy

in topology

collection of sets in which every two sets have the same intersection

thumb|upright|The clock at its original location in May 1979, displaying 17:54 (5:54pm).

in set theory, a subset of a cardinal that intersects every club set

logical principle

partially ordered set such that, for any element, the subset of elements less than it is well-ordered

class of cardinal numbers

set theory construction

discussion of paradoxes of set theory

In set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set. The notion of "negligible" depends on the context, and may mean "of measure zero" (in a measure space), "finite" (when infinite sets are involved), or "countable" (when uncountably infinite sets are involved).

real number uniquely specified by description

viewpoint in the philosophy of mathematics

theorem in combinatorial set theory

in mathematics, notion of limit for sequences of sets

closed cofinal subset of an ordinal

set theory method

subject field of Boolean algebra discussing changes of Boolean variables and functions

Finite sets whose elements are all hereditarily finite sets

first article on transfinite set theory

minimal graph with the same reachability relation as a given graph

extension of ideas in combinatorics to infinite sets