Glossary of set theory

This is a glossary of set theory.

Greek
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Ω:

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∈, =, ⊆, ⊇, ⊃, ⊂, ∪, ∩, ∅:

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∞:

$\alpha^\beta$:

${}^\beta\alpha$:

→:

f x:

f X:

[ ]:

{ }:

⟨ ⟩:

$:

$\:

⌜φ⌝:

⊦:

⊧:

⊩:

≺:

⊥:

0#:

0†:

ℵ:

ב:

ג:

ת:

A
𝔞:

A:

absolute:

AC:

AD:

add: additivity:

additively:

admissible:

AH:

aleph:

almost universal:

amenable:

analytic:

analytical:

antichain:

anti-foundation axiom: An axiom in set theory that allows for the existence of non-well-founded sets, in contrast to the traditional foundation axiom which prohibits such sets.

antinomy:

arithmetic:

arithmetical:

Aronszajn:

atom:

atomic:

axiom: Aczel's anti-foundation axiom states that every accessible pointed directed graph corresponds to a unique set AD+ An extension of the axiom of determinacy Axiom F states that the class of all ordinals is Mahlo Axiom of adjunction Adjoining a set to another set produces a set Axiom of amalgamation The union of all elements of a set is a set. Same as axiom of union Axiom of choice The product of any set of non-empty sets is non-empty Axiom of collection This can mean either the axiom of replacement or the axiom of separation Axiom of comprehension The class of all sets with a given property is a set. Usually contradictory. Axiom of constructibility Any set is constructible, often abbreviated as V=L Axiom of countability Every set is hereditarily countable Axiom of countable choice The product of a countable number of non-empty sets is non-empty Axiom of dependent choice A weak form of the axiom of choice Axiom of determinacy Certain games are determined, in other words one player has a winning strategy Axiom of elementary sets describes the sets with 0, 1, or 2 elements Axiom of empty set The empty set exists Axiom of extensionality or axiom of extent Axiom of finite choice Any product of non-empty finite sets is non-empty Axiom of foundation Same as axiom of regularity Axiom of global choice There is a global choice function Axiom of heredity (any member of a set is a set; used in Ackermann's system.) Axiom of infinity There is an infinite set Axiom of limitation of size A class is a set if and only if it has smaller cardinality than the class of all sets Axiom of pairing Unordered pairs of sets are sets Axiom of power set The powerset of any set is a set Axiom of projective determinacy Certain games given by projective set are determined, in other words one player has a winning strategy Axiom of real determinacy Certain games are determined, in other words one player has a winning strategy Axiom of regularity Sets are well founded Axiom of replacement The image of a set under a function is a set. Same as axiom of substitution Axiom of subsets The powerset of a set is a set. Same as axiom of powersets Axiom of substitution The image of a set under a function is a set Axiom of union The union of all elements of a set is a set Axiom schema of predicative separation Axiom of separation for formulas whose quantifiers are bounded Axiom schema of replacement The image of a set under a function is a set Axiom schema of separation The elements of a set with some property form a set Axiom schema of specification The elements of a set with some property form a set. Same as axiom schema of separation Freiling's axiom of symmetry is equivalent to the negation of the continuum hypothesis Martin's axiom states very roughly that cardinals less than the cardinality of the continuum behave like ℵ0. The proper forcing axiom is a strengthening of Martin's axiom

B
𝔟:

B:

BA:

BACH:

Baire:

basic set theory:

BC:

BD:

Berkeley cardinal:

Bernays:

Berry's paradox:

beth:

Beth:

BG:

BGC:

boldface:

Boolean algebra:

Borel:

bounding number:

BP:

BS: BST:

Burali-Forti:

C
c: 𝔠:

∁:

C:

cac:

Cantor:

Card:

Cartesian product:

cardinal:

cardinality:

categorical:

category:

ccc:

cf:

CH:

chain:

characteristic function: A function that indicates membership of an element in a set, taking the value 1 if the element is in the set and 0 otherwise.

choice function: A function that, given a set of non-empty sets, assigns to each set an element from that set. Fundamental in the formulation of the axiom of choice in set theory.

choice negation: In logic, an operation that negates the principles underlying the axiom of choice, exploring alternative set theories where the axiom does not hold.

choice set: A set constructed from a collection of non-empty sets by selecting one element from each set, related to the concept of a choice function.

cl:

class:

class comprehension schema: A principle in set theory allowing the formation of classes based on properties or conditions that their members satisfy.

club:

coanalytic:

cofinal:

cof:

cofinality:

cofinite: Referring to a set whose complement in a larger set is finite, often used in discussions of topology and set theory.

Cohen:

Col: collapsing algebra:

combinatorial set theory: A branch of set theory focusing on the study of combinatorial properties of sets and their implications for the structure of the mathematical universe.

compact cardinal: A cardinal number that is uncountable and has the property that any collection of sets of that cardinality has a subcollection of the same cardinality with a non-empty intersection.

complement (of a set): The set containing all elements not in the given set, within a larger set considered as the universe.

complete:

Con:

condensation lemma:

constructible:

continuum:

continuum hypothesis: The hypothesis in set theory that there is no set whose cardinality is strictly between that of the integers and the real numbers.

continuum many: An informal way of saying that a set has the cardinality of the continuum, the size of the set of real numbers.

continuum problem: The problem of determining the possible cardinalities of infinite sets, including whether the continuum hypothesis is true.

core:

countable: A set is countable if it is finite or if its elements can be put into a one-to-one correspondence with the natural numbers.

countable antichain condition:

countable cardinal: A cardinal number that represents the size of a countable set, typically the cardinality of the set of natural numbers.

countable chain condition:

countable ordinal: An ordinal number that represents the order type of a well-ordered set that is countable, including all finite ordinals and the first infinite ordinal, $\omega$.

countably infinite: A set that has the same cardinality as the set of natural numbers, meaning its elements can be listed in a sequence without end.

cov(I): covering number:

critical:

CRT:

CTM:

cumulative hierarchy:

D
𝔡:

DC:

Dedekind:

def:

definable:

delta:

denumerable:

dependent choice: See Axiom of dependent choice

determinateness: See Axiom of extensionality

Df:

diagonal argument: Cantor's diagonal argument

diagonalization: A method used in set theory and logic to construct a set or sequence that is not in a given collection by ensuring it differs from each member of the collection in at least one element.

diagonal intersection:

diamond principle:

discrete: A property of a set or space that consists of distinct, separate elements or points, with no intermediate values.

disjoint: Referring to sets that have no element in common, i.e., their intersection is empty.

dom:

DST:

E
E:

Easton's theorem:

EATS:

effectively decidable set: A set for which there exists an algorithm that can determine, for any given element, whether it belongs to the set.

effectively enumerable set: A set whose members can be listed or enumerated by some algorithm, even if the list is potentially infinite.

element: An individual object or member of a set.

elementary:

empty set: The unique set that contains no elements, denoted by $\emptyset$.

empty set axiom: See Axiom of empty set.

enumerable set: A set whose elements can be put into a one-to-one correspondence with the set of natural numbers, making it countable.

enumeration: The process of listing or counting elements in a set, especially for countable sets.

epsilon:

equinumerous: Having the same cardinal number or number of elements, used to describe two sets that can be put into a one-to-one correspondence.

equipollent: Synonym of equinumerous

equivalence class: A subset within a set, defined by an equivalence relation, where every element in the subset is equivalent to each other under that relation.

Erdos: Erdős:

ethereal cardinal:

Euler diagram:

extender:

extendible cardinal:

extension:

extensional:

F
F:

Feferman–Schütte ordinal:

filter:

finite intersection property: FIP:

first:

Fodor:

forcing:

formula:

foundation axiom: See Axiom of foundation

Fraenkel:

G
𝖌:

G:

gamma number:

GCH:

generalized continuum hypothesis:

generic:

gimel:

global choice:

global well-ordering: Another name for the axiom of global choice

greatest lower bound: The largest value that serves as a lower bound for a set in a partially ordered set, also known as the infimum.

Godel: Gödel:

H
𝔥:

H:

Hκ: H(κ):

Hartogs:

Hausdorff:

HC:

hereditarily:

Hessenberg:

HF:

Hilbert:

HS:

HOD:

huge cardinal:

hyperarithmetic:

hyperinaccessible: hyper-inaccessible:

hyper-Mahlo:

hyperset: A set that can contain itself as a member or is defined in terms of a circular or self-referential structure, used in the study of non-well-founded set theories.

hyperverse:

I
𝔦:

I0, I1, I2, I3:

ideal:

iff:

improper:
 * See proper, below.

inaccessible cardinal:

indecomposable ordinal:

independence number:

indescribable cardinal:

individual:

indiscernible:

inductive:

infinity axiom: See Axiom of infinity.

inner model: A model of set theory that is constructed within Zermelo-Fraenkel set theory and contains all ordinals of the universe, serving to explore properties of larger set-theoretic universes from a contained perspective.

ineffable cardinal:

inner model:

Int:

integers: The set of whole numbers including positive, negative, and zero, denoted by $ \mathbb{Z} $.

internal:

intersection: The set containing all elements that are members of two or more sets, denoted by $A \cap B$ for sets $A$ and $B$.

iterative conception of set: A philosophical and mathematical notion that sets are formed by iteratively collecting together objects into a new object, a set, which can then itself be included in further sets.

J
j:

J:

Jensen:

join: In logic and mathematics, particularly in lattice theory, the join of a set of elements is the least upper bound or supremum of those elements, representing their union in the context of set operations or the least element that is greater than or equal to each of them in a partial order.

Jónsson:

K
Kelley:

KH:

kind:

KM:

Kleene–Brouwer ordering:

Kleene hierarchy: A classification of sets of natural numbers or strings based on the complexity of the predicates defining them, using Kleene's arithmetical hierarchy in recursion theory.

König's lemma: A result in graph theory and combinatorics stating that every infinite, finitely branching tree has an infinite path, used in proofs of various mathematical and logical theorems. It is equivalent to the axiom of dependent choice.

König's paradox: A paradox in set theory and combinatorics that arises from incorrect assumptions about infinite sets and their cardinalities, related to König's theorem on the sums and products of cardinals.

KP:

Kripke:

Kuratowski:

Kurepa:

L
L:

large cardinal:

lattice: A partially ordered set in which any two elements have a unique supremum (least upper bound) and an infimum (greatest lower bound), used in various areas of mathematics and logic.

Laver:

least upper bound: The smallest element in a partially ordered set that is greater than or equal to every element in a subset of that set, also known as the supremum.

Lebesgue:

LEM:

Lévy:

lightface:

limit:

limitation-of-size conception of set: A conception that defines sets in such a way as to avoid certain paradoxes by excluding collections that are too large to be sets.

limited:

LM:

local:

LOTS:

Löwenheim:

lower bound: An element of a partially ordered set that is less than or equal to every element of a given subset of the set, providing a minimum standard or limit for comparison.

LST:

M
m:

𝔪:

M:

MA:

MAD:

Mac Lane:

Mahlo:

Martin:

meager: meagre:

measure:

measurable cardinal:

meet: In lattice theory, the operation that combines two elements to produce their greatest lower bound, analogous to intersection in set theory.

member: An individual element of a set.

membership: The relation between an element and a set in which the element is included within the set.

mice:

Milner–Rado paradox:

MK:

MM:

morass:

Morse:

Mostowski:

mouse:

multiplicative axiom:

multiset: A generalization of a set that allows multiple occurrences of its elements, often used in mathematics and computer science to model collections with repetitions.

N
N:

naïve comprehension schema: An unrestricted principle in set theory allowing the formation of sets based on any property or condition, leading to paradoxes such as Russell's paradox in naïve set theory.

naive set theory:

natural:

NCF:

no-classes theory:

non:

nonstat: nonstationary:

normal:

NS:

null:

number class:

O
OCA:

OD:

Omega logic:

On:

order type: A concept in set theory and logic that categorizes well-ordered sets by their structure, such that two sets have the same order type if there is a bijective function between them that preserves order.

ordinal:

ot:

P
𝔭:

P:

pairing function:

pairwise disjoint: A property of a collection of sets where each pair of sets in the collection has no elements in common.

pantachie: pantachy:

paradox:

paradox of denotation: A paradox that uses definite descriptions in an essential way, such as Berry's paradox, König's paradox, and Richard's paradox.

partial order:

partition: A division of a set into disjoint subsets whose union is the entire set, with no element being left out.

partition cardinal:

PCF:

PD:

perfect set:

permutation: A rearrangement of the elements of a set or sequence, where the structure of the set changes but the elements do not.

permutation model:

PFA:

PM:

po:

poset:

positive set theory: A variant of set theory that includes a universal set and possibly other non-standard axioms, focusing on what can be constructed or defined positively.

Polish space:

pow:

power:

power set: powerset:

pre-ordering: A relation that is reflexive and transitive but not necessarily antisymmetric, allowing for the comparison of elements in a set.

primitive recursive set: A set whose characteristic function is a primitive recursive function, indicating that membership in the set can be decided by a computable process.

projective:

proper:

PSP:

pure set: A term for hereditary sets, which are sets that have only other sets as elements, that is, without any urelements.

pure set theory: A set theory that deals only with pure sets, also known as hereditary sets

Q
Q:

QPD:

quantifier:

Quasi-projective determinacy:

R
𝔯:

R:

Ramsey:

ran:

rank:

recursive set: A set for which membership can be decided by a recursive procedure or algorithm, also known as a decidable or computable set.

recursively enumerable set: A set for which there exists a Turing machine that will list all members of the set, possibly without halting if the set is infinite; also called "semi-decidable set" or "Turing recognizable set".

reflecting cardinal:

reflection principle:

regressive:

regular:

Reinhardt cardinal:

relation:

relative complement: The set of elements that are in one set but not in another, often denoted as $A \setminus B$ for sets $A$ and $B$.

Richard:

RO:

Rowbottom:

rud:

rudimentary:

rudimentary set theory:

Russell:

Russell set:

S
𝔰:

Satisfaction relation:

SBH:

SCH:

SCS:

Scott:

second:

semi-decidable set: A set for which membership can be determined by a computational process that halts and accepts if the element is a member, but may not halt if the element is not a member.

sentence:

separating set:

separation axiom: In set theory, sometimes refers to the Axiom schema of separation; not to be confused with the Separation axiom from topology.

separative:

set:

set-theoretic:

singleton: A set containing exactly one element; its significance lies in its role in the definition of functions and in the formulation of mathematical and logical concepts.

SFIP:

SH:

Shelah:

shrewd cardinal:

Sierpinski: Sierpiński:

Silver:

simply infinite set:

singular:

SIS:

Skolem:

small:

SOCA:

Solovay:

special:

square:

standard model:

stationary set:

stratified: A formula of set theory is stratified if and only if there is a function $\sigma$ which sends each variable appearing in $\phi$ (considered as an item of syntax) to a natural number (this works equally well if all integers are used) in such a way that any atomic formula $x \in y$ appearing in $\phi$ satisfies $\sigma(x)+1 = \sigma(y)$ and any atomic formula $x = y$ appearing in $\phi$ satisfies $\sigma(x) = \sigma(y)$.

strict ordering: An ordering relation that is transitive and irreflexive, implying that no element is considered to be strictly before or after itself, and that the relation holds transitively.

strong:

strongly:

subset: A set whose members are all contained within another set, without necessarily being identical to it.

subtle cardinal:

successor:

such that:

sunflower:

Souslin: Suslin:

A Suslin representation of a set of reals is a tree whose projection is that set of reals A Suslin scheme is a function with domain the finite sequences of positive integers A Suslin set is a set that is the image of a tree under a certain projection A Suslin space is the image of a Polish space under a continuous mapping A Suslin subset is a subset that is the image of a tree under a certain projection The Suslin theorem about analytic sets states that a set that is analytic and coanalytic is Borel A Suslin tree is a tree of height ω1 such that every branch and every antichain is at most countable.

supercompact:

super transitive: supertransitive:

symmetric difference: The set operation that yields the elements present in either of two sets but not in their intersection, effectively the elements unique to each set.

symmetric model:

T
𝔱:

T:

tall cardinal:

Tarski:

TC:

total order:

totally indescribable:

transfinite:

transitive:

tree:

tuple: An ordered list of elements, with a fixed number of components, used in mathematics and computer science to describe ordered collections of objects.

Turing recognizable set: A set for which there exists a Turing machine that halts and accepts on any input in the set, but may either halt and reject or run indefinitely on inputs not in the set.

type class:

U
𝔲:

Ulam:

Ult:

ultrafilter:

ultrapower:

ultraproduct:

unfoldable cardinal:

uniformity:

uniformization:

union: An operation in set theory that combines the elements of two or more sets to form a single set containing all the elements of the original sets, without duplication.

universal: universe:

unordered pair: A set of two elements where the order of the elements does not matter, distinguishing it from an ordered pair where the sequence of elements is significant. The axiom of pairing asserts that for any two objects, the unordered pair containing those objects exists.

upper bound: In mathematics, an element that is greater than or equal to every element of a given set, used in the discussion of intervals, sequences, and functions.

upward Löwenheim–Skolem theorem: A theorem in model theory stating that if a countable first-order theory has an infinite model, then it has models of all larger cardinalities, demonstrating the scalability of models in first-order logic. (See Löwenheim–Skolem theorem)

urelement:

V
V:

V=L:

Veblen:

Venn diagram:

von Neumann:

Vopenka: Vopěnka:

W
weakly:

well-founded:

well-ordering:

well-ordering principle:

well-ordering theorem:

Wf:

Woodin:

XYZ
Z:

ZC:

Zermelo:

ZF:

ZFA:

ZFC:

zero function: A mathematical function that always returns the value zero, regardless of the input, often used in discussions of functions, calculus, and algebra.

ZF-P:

Zorn: