Set theory: vocabulary and definitions (130 words)

This page gathers the 130 dictionary entries belonging to the domain “Set theory”, as labelled by Wiktionary. Each word leads to its full entry: definitions, etymology, pronunciation, examples. Page 1 of 2: from “abstract polyhedron” to “saturated”.

abstract polyhedron noun A type of abstract polytope where the connections between its points correspond to the faces, edges, and vertices of a polyhedron.
abstract polytope noun A partially ordered set that captures the dyadic properties of a normal polytope without specifying its geometric properties.
ac noun Initialism of axiom of choice.
aleph number noun Any of a sequence of numbers used to represent the cardinality of infinite sets, denoted by the Hebrew letter aleph.
aleph-one noun The second of the transfinite cardinal numbers; according to the continuum hypothesis, it corresponds to the number of real numbers.
algebra noun A collection of subsets of a given set, such that this collection contains the empty set, and the collection is closed under unions and…
amorphous adj Infinite and not the disjoint union of two infinite subsets.
antichain noun A subset, A, of a partially ordered set, (P, ≤), such that no two elements of A are comparable with respect to ≤.
antisymmetric adj Having the property that, for any two distinct elements of S, at least one is not related to the other via R; equivalently, having the…
antisymmetry noun The condition of being antisymmetric.
asymmetric adj Of a relation R on a set S: having the property that for any two elements of S (not necessarily distinct), at least one is not related to…
atom noun An element of a set that is not itself a set; an urelement.
axiom of choice noun One of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version…
axiom of countable choice noun A weaker form of the axiom of choice that states that every countable collection of nonempty sets must have a choice function…
axiom of extensionality noun One of the axioms in axiomatic set theory, equivalent to the statement that two sets are equal if and only if they contain the same…
axiom of pairing noun One of the axioms in axiomatic set theory, equivalent to the statement that if two sets exist, there exists a set with those two sets as…
axiom of regularity noun One of the axioms in axiomatic set theory, equivalent to the statement that every non-empty set contains a member that is disjoint from…
axiom of union noun One of the axioms in axiomatic set theory, equivalent to the statement that a union exists for any set, containing exactly all the elements…
axiom schema of abstraction noun The axiom schema stating that for any clearly stated condition Sx, there exists a set B whose elements are exactly those objects which…
axiom schema of separation noun The axiom schema stating that for every set A and every condition Sx, there exists a set B whose elements are exactly those members of A…
belong verb (followed by to) To be an element of (a set). The symbol ∈ means belongs to.
binary relation noun A subset of the Cartesian product A×A (the set of ordered pairs (a, b) of elements of A, alternatively written as A²).
bounded adj That contains a least element, a, and a greatest element, b, such that for all x ∈ X, a ≤ x ≤ b.
Cantorian adj Smaller than its own power set; satisfying Cantor's theorem.
cardinality noun The number of elements a given set contains.
Cartesian product noun The set of all possible ordered pairs of elements, the being first from X, the second from Y, written X×Y. Formally, the set…
Cartesian square noun For a set X, the Cartesian product X² = X × X.
chain noun A totally ordered set, especially a totally ordered subset of a poset.
choice noun Ellipsis of axiom of choice.
choice function noun A function whose domain is a family of nonempty sets, and which selects a member from each of those sets as its value.
class noun A collection of sets definable by a shared property, especially one which is not itself a set (in which case the class is called proper).
clubsuit noun A family of combinatorial principles that is a weaker version of the corresponding diamond principle.
codisjoint adj Of two or more sets, having an union equal to the universal set.
collection noun A set of sets; used because such a thing is in general too large to comply with the formal definition of a set.
complement noun Given two sets, the set containing one set's elements that are not members of the other set (whether a relative complement or an absolute…
computable adj Of a countably infinite set, having a computable indicator function.
connex adj Such that, for all x and y in X, and for a binary relation R, either or both of xRy and yRx hold(s).
continuum hypothesis noun The hypothesis which states that any infinite subset of ℝ must have the cardinality of either the set of natural numbers or of ℝ itself.
correspondence noun A relation.
countably infinite adj That is both countable and infinite; having the same cardinality as the set of natural numbers; formally, such that a bijection exists from…
creative adj A type of set of natural numbers, related to mathematical logic.
direct product noun The set of all possible tuples whose elements are elements of given, separately specified, sets.
directed adj Having the properties of a directed set.
directed set noun A set equipped with a preorder such that for any two elements a and b there is an element c such that a ≤ c and b ≤ c.
disjoint adj Of two or more sets, having no members in common; having an intersection equal to the empty set.
dismantlable adj The property of an ordered set such that its elements can be listed in an order such that, for every element, the element is irreducible…
domain noun The set of input (argument) values for which a function is defined.
element noun One of the objects in a set.
epsilon number noun Any (necessarily transfinite) ordinal number α such that ω^α = α; (by generalisation) any surreal number that is a fixed point of the…
equivalence class noun Any one of the subsets into which an equivalence relation partitions a set, each of these subsets containing all the elements of the set…
extensionality noun The principle, codified in the axiom of extensionality, that sets are equal if and only if they contain the same elements.
family noun A collection of sets, especially of subsets of a given set.
filtration noun A totally ordered collection of subsets.
forcing noun A technique used to prove the consistency of certain axioms in set theory. See forcing (mathematics).
generalized continuum hypothesis noun The hypothesis that, for each ordinal α, there is no cardinal number strictly between א_α and 2^(א_α), i.e. 2^(א_α)=א_(α+1).
Goguen category noun A fuzzy kind of category in which membership is determined not in binary (true-or-false) fashion but by a more general interval.
Hartogs number noun For a given set X, the cardinality of the least ordinal number α such that there is no injection from α into X.
Hasse diagram noun A diagram which represents a finite poset, in which nodes are elements of the poset and arrows represent the order relation between…
Hausdorff gap noun A pair of collections of integer sequences such that there is no integer sequence lying between the two.
homogeneous adj Holding between a set and itself; being an endorelation.
ideal noun A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also…
independence number noun The smallest cardinality of a maximal independent family of subsets of the natural numbers, usually denoted by lowercase Fraktur letter i.
infinite adj Having infinitely many elements.
injection noun A function that maps distinct x in the domain to distinct y in the codomain; formally, a f: X → Y such that f(a) = f(b) implies a = b for…
intersection noun The set containing all the elements that are common to two or more sets.
kernel noun The set of pairs of elements in the domain of f which are mapped to the same value; the equivalence relation x≡y⟺f(x)=f(y).
Kurepa tree noun A tree (T, <) of height ω₁, each of whose levels is at most countable, and has at least ℵ₂ branches.
L-fuzzy adj Relating to a generalized variant of the notion of fuzzy sets, with membership functions taking values in a (fixed or variable) algebra or…
left total adj Such that every element of the domain is related to at least one element of the codomain: such that ∀a∈A;;∃b∈B:(a,b)∈R
limit cardinal noun A cardinal number that cannot be reached from another by repeated successor operations (a weak limit cardinal) or by repeated power set…
meager adj Of a set: such that, considered as a subset of a (usually larger) topological space, it is in a precise sense small or negligible.
meagre adj Of a set: such that, considered as a subset of a (usually larger) topological space, it is in a precise sense small or negligible.
member noun An element of a set.
memberless adj Without members; empty or null.
mouse noun A small model of (a fragment of) Zermelo-Fraenkel set theory with desirable properties (depending on the context).
normal adj Which is strictly monotonically increasing and continuous with respect to the order topology.
null set noun The empty set.
omega noun A transfinite ordinal number referring to the next position after ordering a countably infinite set.
omegath adj In the position known as omega (a transfinite ordinal number referring to the next position after ordering a countably infinite set).
one num The cardinality of the smallest nonempty set.
order noun The number of elements contained within (the given object); formally, the cardinality (of the given object).
order type noun In the context of sets equipped with an order (especially, the context of totally ordered sets), the characteristic of being a member of…
ordered pair noun An object containing exactly two elements in a fixed order, so that, when the elements are different, exchanging them gives a different…
ordinal number noun Such a number generalised to correspond to any cardinal number (the size of some set); formally, the order type of some well-ordered set of…
pairwise disjoint adj Let A_λ_(λ∈Λ) be any collection of sets indexed by a set Λ. We call the indexed collection pairwise disjoint if for any two distinct…
partial order noun (informal) An ordering of the elements of a collection that behaves like that of the natural numbers by size, except that some elements may…
partially ordered set noun A set that has a given, elsewhere specified partial order.
partition noun A collection of non-empty, disjoint subsets of a set whose union is the set itself (i.e. all elements of the set are contained in exactly…
pointclass noun A collection of sets of points (ordinarily understood to be elements of some perfect Polish space), characterized by some sort of…
power noun Cardinality.
preorder noun A binary relation that is reflexive and transitive.
productive adj A type of set of natural numbers, related to mathematical logic; a set S is productive if there exists a total recursive function f such…
proper adj Not being a set.
proper inclusion noun The state of being a proper subset; a subset relation in which a set A is a subset of a set B but A is not equal to B.
properly included adj Being a proper subset; having the property that every element of a set A is also an element of a set B, and A is not equal to B.
quasireflexive adj Of a relation ~ on a set S: such that every element that is related to some element is also related to itself, i.e. ∀x,y∈S: x~y ⇒ x~x ∧ y~y.
ramified forcing noun The original form of forcing, starting with a model M of set theory in which the axiom of constructibility, V = L, holds, and then building…
relation noun A set of ordered tuples.
right total adj Of a binary relation, to have every element of the right set occur at least once: ∀b∈B∃a∈A:(a,b)∈R⊆A×B
saturated adj Equal to a union of equivalence classes of ≡; such that if x∈S and x≡y then y∈S.

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