Glossary of set theory

This is a glossary of set theory.

Greek

α: Often used for an ordinal
β: 1. βX is the Stone–Čech compactification of X; 2. An ordinal
γ: A gamma number, an ordinal of the form ω^α
Γ: The Gamma function of ordinals. In particular Γ₀ is the Feferman–Schütte ordinal.
δ: 1. A delta number is an ordinal of the form ω^{ω^α}; 2. A limit ordinal
Δ (Greek capital delta, not to be confused with a triangle ∆): 1. A set of formulas in the Lévy hierarchy; 2. A delta system
ε: An epsilon number, an ordinal with ω^ε=ε
η: 1. The order type of the rational numbers; 2. An eta set, a type of ordered set; 3. η_α is an Erdős cardinal
θ: The order type of the real numbers
Θ: The supremum of the ordinals that are the image of a function from ^ωω (usually in models where the axiom of choice is not assumed)
κ: 1. Often used for a cardinal, especially the critical point of an elementary embedding; 2. The Erdős cardinal $κ (α)$ is the smallest cardinal such that $κ (α) \to (α) < ω$
λ: 1. Often used for a cardinal; 2. The order type of the real numbers
μ: A measure
Π: 1. A product of cardinals; 2. A set of formulas in the Lévy hierarchy
ρ: The rank of a set
σ: countable, as in σ-compact, σ-complete and so on
Σ: 1. A sum of cardinals; 2. A set of formulas in the Lévy hierarchy
φ: A Veblen function
ω: 1. The smallest infinite ordinal; 2. ω_α is an alternative name for ℵ_α, used when it is considered as an ordinal number rather than a cardinal number; 3. An ω-huge cardinal is a large cardinal related to the I₁ rank-into-rank axiom
Ω: 1. The class of all ordinals, related to Cantor's absolute; 2. Ω-logic is a form of logic introduced by Hugh Woodin

!$@

∈, =, ⊆, ⊇, ⊃, ⊂, ∪, ∩, ∅: Standard set theory symbols with their usual meanings (is a member of, equals, is a subset of, is a superset of, is a proper superset of, is a proper subset of, union, intersection, empty set)
∧ ∨ → ↔ ¬ ∀ ∃: Standard logical symbols with their usual meanings (and, or, implies, is equivalent to, not, for all, there exists)
≡: An equivalence relation
⨡: f⨡X is now the restriction of a function or relation f to some set X, though its original meaning was the corestriction
↿: f↿X is the restriction of a function or relation f to some set X
∆ (A triangle, not to be confused with the Greek letter Δ): 1. The symmetric difference of two sets; 2. A diagonal intersection
◊: The diamond principle
♣: A clubsuit principle
□: The square principle
∘: The composition of functions
⁀: s⁀x is the extension of a sequence s by x
+: 1. Addition of ordinals; 2. Addition of cardinals; 3. α⁺ is the smallest cardinal greater than α; 4. B⁺ is the poset of nonzero elements of a Boolean algebra B; 5. The inclusive or operation in a Boolean algebra. (In ring theory it is used for the exclusive or operation)
~: 1. The difference of two sets: x~y is the set of elements of x not in y.; 2. An equivalence relation
\: The difference of two sets: x\y is the set of elements of x not in y.
−: The difference of two sets: x−y is the set of elements of x not in y.
≈: Has the same cardinality as
×: A product of sets
/: A quotient of a set by an equivalence relation
⋅: 1. x⋅y is the ordinal product of two ordinals; 2. x⋅y is the cardinal product of two cardinals
*: An operation that takes a forcing poset and a name for a forcing poset and produces a new forcing poset.
∞: The class of all ordinals, or at least something larger than all ordinals
$\alpha ^{\beta }$: 1. Cardinal exponentiation; 2. Ordinal exponentiation
${}^{\beta }\alpha$: 1. The set of functions from β to α
→: 1. Implies; 2. f:X→Y means f is a function from X to Y.; 3. The ordinary partition symbol, where κ→(λ)ⁿ
_m means that for every coloring of the n-element subsets of κ with m colors there is a subset of size λ all of whose n-element subsets are the same color.
f ' x: If there is a unique y such that ⟨x,y⟩ is in f then f ' x is y, otherwise it is the empty set. So if f is a function and x is in its domain, then f ' x is f(x).
f “ X: f “ X is the image of a set X by f. If f is a function whose domain contains X this is {f(x):x∈X}
[ ]: 1. M[G] is the smallest model of ZF containing G and all elements of M.; 2. [α]^β is the set of all subsets of a set α of cardinality β, or of an ordered set α of order type β; 3. [x] is the equivalence class of x
{ }: 1. {a, b, ...} is the set with elements a, b, ...; 2. {x : φ(x)} is the set of x such that φ(x)
⟨ ⟩: ⟨a,b⟩ is an ordered pair, and similarly for ordered n-tuples
|X|: The cardinality of a set X
||φ||: The value of a formula φ in some Boolean algebra
⌜φ⌝: ⌜φ⌝ (Quine quotes, unicode U+231C, U+231D) is the Gödel number of a formula φ
⊦: A⊦φ means that the formula φ follows from the theory A
⊧: A⊧φ means that the formula φ holds in the model A
⊩: The forcing relation
≺: An elementary embedding
⊥: The false symbol; p⊥q means that p and q are incompatible elements of a partial order
0^#: zero sharp, the set of true formulas about indiscernibles and order-indiscernibles in the constructible universe
0^†: zero dagger, a certain set of true formulas
ℵ: The Hebrew letter aleph, which indexes the aleph numbers or infinite cardinals ℵ_α
ב: The Hebrew letter beth, which indexes the beth numbers ב_α
$\gimel$: A serif form of the Hebrew letter gimel, representing the gimel function $\gimel (\kappa )=\kappa ^{{\text{cf }}\kappa }$
ת: The Hebrew letter Taw, used by Cantor for the class of all cardinal numbers

A

𝔞: The almost disjointness number, the least size of a maximal almost disjoint family of infinite subsets of ω
A: The Suslin operation
absolute: 1. A statement is called absolute if its truth in some model implies its truth in certain related models; 2. Cantor's absolute is a somewhat unclear concept sometimes used to mean the class of all sets; 3. Cantor's Absolute Infinite Ω is a somewhat unclear concept related to the class of all ordinals
AC: 1. AC is the Axiom of choice; 2. AC_ω is the Axiom of countable choice
AD: The axiom of determinacy
add
additivity: The additivity add(I) of I is the smallest number of sets of I with union not in I
additively: An ordinal is called additively indecomposable if it is not the sum of a finite number of smaller ordinals. These are the same as gamma numbers or powers of ω.
admissible: An admissible set is a model of Kripke–Platek set theory, and an admissible ordinal is an ordinal α such that L_α is an admissible set
AH: Aleph hypothesis, a form of the generalized continuum hypothesis
aleph: 1. The Hebrew letter ℵ; 2. An infinite cardinal; 3. The aleph function taking ordinals to infinite cardinals; 4. The aleph hypothesis is a form of the generalized continuum hypothesis
almost universal: A class is called almost universal if every subset of it is contained in some member of it
amenable: An amenable set is a set that is a model of Kripke–Platek set theory without the axiom of collection
analytic: An analytic set is the continuous image of a Polish space. (This is not the same as an analytical set)
analytical: The analytical hierarchy is a hierarchy of subsets of an effective Polish space (such as ω). They are definable by a second-order formula without parameters, and an analytical set is a set in the analytical hierarchy. (This is not the same as an analytic set)
antichain: An antichain is a set of pairwise incompatible elements of a poset
antinomy: paradox
arithmetic: The ordinal arithmetic is arithmetic on ordinal numbers; The cardinal arithmetic is arithmetic on cardinal numbers
arithmetical: The arithmetical hierarchy is a hierarchy of subsets of a Polish space that can be defined by first-order formulas
Aronszajn: 1. Nachman Aronszajn; 2. An Aronszajn tree is an uncountable tree such that all branches and levels are countable. More generally a κ-Aronszajn tree is a tree of cardinality κ such that all branches and levels have cardinality less than κ
atom: 1. An urelement, something that is not a set but allowed to be an element of a set; 2. An element of a poset such that any two elements smaller than it are compatible.; 3. A set of positive measure such that every measurable subset has the same measure or measure 0
atomic: An atomic formula (in set theory) is one of the form x=y or x∈y
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: The bounding number, the least size of an unbounded family of sequences of natural numbers
B: A Boolean algebra
BA: Baumgartner's axiom, one of three axioms introduced by Baumgartner.
BACH: Baumgartner's axiom plus the continuum hypothesis.
Baire: 1. René-Louis Baire; 2. A subset of a topological space has the Baire property if it differs from an open set by a meager set; 3. The Baire space is a topological space whose points are sequences of natural numbers; 4. A Baire space is a topological space such that every intersection of a countable collection of open dense sets is dense
basic set theory: 1. Naive set theory; 2. A weak set theory, given by Kripke–Platek set theory without the axiom of collection
BC: Berkeley cardinal
BD: Borel determinacy
Berkeley cardinal: A Berkeley cardinal is a cardinal κ in a model of ZF such that for every transitive set M that includes κ, there is a nontrivial elementary embedding of M into M with critical point below κ.
Bernays: 1. Paul Bernays; 2. Bernays–Gödel set theory is a set theory with classes
Berry's paradox: Berry's paradox considers the smallest positive integer not definable in ten words
Berkeley cardinal: A Berkeley cardinal is a cardinal κ in a model of ZF such that for every transitive set M that includes κ, there is a nontrivial elementary embedding of M into M with critical point below κ.
beth: 1. The Hebrew letter ב; 2. A beth number ב_α
Beth: Evert Willem Beth
BG: Bernays–Gödel set theory without the axiom of choice
BGC: Bernays–Gödel set theory with the axiom of choice
boldface: The boldface hierarchy is a hierarchy of subsets of a Polish space, definable by second-order formulas with parameters (as opposed to the lightface hierarchy which does not allow parameters). It includes the Borel sets, analytic sets, and projective sets
Boolean algebra: A Boolean algebra is a commutative ring such that all elements satisfy x²=x
Borel: 1. Émile Borel; 2. A Borel set is a set in the smallest sigma algebra containing the open sets
bounding number: The bounding number is the least size of an unbounded family of sequences of natural numbers
BP: Baire property
BS
BST: Basic set theory
Burali-Forti: 1. Cesare Burali-Forti; 2. The Burali-Forti paradox states that the ordinal numbers do not form a set

C

c
𝔠: The cardinality of the continuum
∁: Complement of a set
C: The Cantor set
cac: countable antichain condition (same as the countable chain condition)
Cantor: 1. Georg Cantor; 2. The Cantor normal form of an ordinal is its base ω expansion.; 3. Cantor's paradox says that the powerset of a set is larger than the set, which gives a contradiction when applied to the universal set.; 4. The Cantor set, a perfect nowhere dense subset of the real line; 5. Cantor's absolute infinite Ω is something to do with the class of all ordinals; 6. Cantor's absolute is a somewhat unclear concept sometimes used to mean the class of all sets; 7. Cantor's theorem states that the powerset operation increases cardinalities
Card: The cardinality of a set
cardinal: 1. A cardinal number is an ordinal with more elements than any smaller ordinal
cardinality: The number of elements of a set
categorical: 1. A theory is called categorical if all models are isomorphic. This definition is no longer used much, as first-order theories with infinite models are never categorical.; 2. A theory is called k-categorical if all models of cardinality κ are isomorphic
category: 1. A set of first category is the same as a meager set: a set that is the union of a countable number of nowhere-dense sets, and a set of second category is a set that is not of first category.; 2. A category in the sense of category theory.
ccc: countable chain condition
cf: The cofinality of an ordinal
CH: The continuum hypothesis
chain: A linearly ordered subset (of a poset)
cl: Abbreviation for "closure of" (a set under some collection of operations)
class: 1. A class is a collection of sets; 2. First class ordinals are finite ordinals, and second class ordinals are countable infinite ordinals
club: A contraction of "closed unbounded"; 1. A club set is a closed unbounded subset, often of an ordinal; 2. The club filter is the filter of all subsets containing a club set; 3. Clubsuit is a combinatorial principle similar to but weaker than the diamond principle
coanalytic: A coanalytic set is the complement of an analytic set
cofinal: A subset of a poset is called cofinal if every element of the poset is at most some element of the subset.
cof: confinality
cofinality: 1. The cofinality of a poset (especially an ordinal or cardinal) is the smallest cardinality of a cofinal subset; 2. The cofinality cof(I) of an ideal I of subsets of a set X is the smallest cardinality of a subset B of I such that every element of I is a subset of something in B.
Cohen: 1. Paul Cohen; 2. Cohen forcing is a method for constructing models of ZFC; 3. A Cohen algebra is a Boolean algebra whose completion is free
Col
collapsing algebra: A collapsing algebra Col(κ,λ) collapses cardinals between λ and κ
complete: 1. "Complete set" is an old term for "transitive set"; 2. A theory is called complete if it assigns a truth value (true or false) to every statement of its language; 3. An ideal is called κ-complete if it is closed under the union of less than κ elements; 4. A measure is called κ-complete if the union of less than κ measure 0 sets has measure 0; 5. A linear order is called complete if every nonempty bounded subset has a least upper bound
Con: Con(T) for a theory T means T is consistent
condensation lemma: Gödel's condensation lemma says that an elementary submodel of an element L_α of the constructible hierarchy is isomorphic to an element L_γ of the constructible hierarchy
constructible: A set is called constructible if it is in the constructible universe.
continuum: The continuum is the real line or its cardinality
core: A core model is a special sort of inner model generalizing the constructible universe
countable antichain condition: A term used for the countable chain condition by authors who think terminology should be logical
countable chain condition: The countable chain condition (ccc) for a poset states that every antichain is countable
cov(I)
covering number: The covering number cov(I) of an ideal I of subsets of X is the smallest number of sets in I whose union is X.
critical: 1. The critical point κ of an elementary embedding j is the smallest ordinal κ with j(κ) > κ; 2. A critical number of a function j is an ordinal κ with j(κ) = κ. This is almost the opposite of the first meaning.
CRT: The critical point of something
CTM: Countable transitive model
cumulative hierarchy: A cumulative hierarchy is a sequence of sets indexed by ordinals that satisfies certain conditions and whose union is used as a model of set theory

D

𝔡: The dominating number of a poset
DC: The axiom of dependent choice
def: The set of definable subsets of a set
definable: A subset of a set is called definable set if it is the collection of elements satisfying a sentence in some given language
delta: 1. A delta number is an ordinal of the form ω^{ω^α}; 2. A delta system, also called a sunflower, is a collection of sets such that any two distinct sets have intersection X for some fixed set X
denumerable: countable and infinite
Df: The set of definable subsets of a set
diagonal intersection: If $\displaystyle \langle X_{\alpha }\mid \alpha <\delta \rangle$ is a sequence of subsets of an ordinal $\displaystyle \delta$ , then the diagonal intersection $\displaystyle \Delta _{\alpha <\delta }X_{\alpha },$ is $\displaystyle \{\beta <\delta \mid \beta \in \bigcap _{\alpha <\beta }X_{\alpha }\}.$
diamond principle: Jensen's diamond principle states that there are sets A_α⊆α for α<ω₁ such that for any subset A of ω₁ the set of α with A∩α = A_α is stationary in ω₁.
dom: The domain of a function
DST: Descriptive set theory

E

E: E(X) is the membership relation of the set X
Easton's theorem: Easton's theorem describes the possible behavior of the powerset function on regular cardinals
EATS: The statement "every Aronszajn tree is special"
elementary: An elementary embedding is a function preserving all properties describable in the language of set theory
epsilon: 1. An epsilon number is an ordinal α such that α=ω^α; 2. Epsilon zero (ε₀) is the smallest epsilon number
Erdos
Erdős: 1. Paul Erdős; 2. An Erdős cardinal is a large cardinal satisfying a certain partition condition. (They are also called partition cardinals.); 3. The Erdős–Rado theorem extends Ramsey's theorem to infinite cardinals
ethereal cardinal: An ethereal cardinal is a type of large cardinal similar in strength to subtle cardinals
extender: An extender is a system of ultrafilters encoding an elementary embedding
extendible cardinal: A cardinal κ is called extendible if for all η there is a nontrivial elementary embedding of V_κ+η into some V_λ with critical point κ
extension: 1. If R is a relation on a class then the extension of an element y is the class of x such that xRy; 2. An extension of a model is a larger model containing it
extensional: 1. A relation R on a class is called extensional if every element y of the class is determined by its extension; 2. A class is called extensional if the relation ∈ on the class is extensional

F

F: An F_σ is a union of a countable number of closed sets
Feferman–Schütte ordinal: The Feferman–Schütte ordinal Γ₀ is in some sense the smallest impredicative ordinal
filter: A filter is a non-empty subset of a poset that is downward-directed and upwards-closed
finite intersection property
FIP: The finite intersection property, abbreviated FIP, says that the intersection of any finite number of elements of a set is non-empty
first: 1. A set of first category is the same as a meager set: one that is the union of a countable number of nowhere-dense sets.; 2. An ordinal of the first class is a finite ordinal; 3. An ordinal of the first kind is a successor ordinal; 4. First-order logic allows quantification over elements of a model, but not over subsets
Fodor: 1. Géza Fodor; 2. Fodor's lemma states that a regressive function on a regular uncountable cardinal is constant on a stationary subset.
forcing: Forcing (mathematics) is a method of adjoining a generic filter G of a poset P to a model of set theory M to obtain a new model M[G]
formula: Something formed from atomic formulas x=y, x∈y using ∀∃∧∨¬
Fraenkel: Abraham Fraenkel

G

𝖌: The groupwise density number
G: 1. A generic ultrafilter; 2. A G_δ is a countable intersection of open sets
gamma number: A gamma number is an ordinal of the form ω^α
GCH: Generalized continuum hypothesis
generalized continuum hypothesis: The generalized continuum hypothesis states that 2^ℵ_α = ℵ_α+1
generic: 1. A generic filter of a poset P is a filter that intersects all dense subsets of P that are contained in some model M.; 2. A generic extension of a model M is a model M[G] for some generic filter G.
gimel: 1. The Hebrew letter gimel $\gimel$; 2. The gimel function $\gimel (\kappa )=\kappa ^{{\text{cf}}(\kappa )}$; 3. The gimel hypothesis states that $\gimel (\kappa )=\max(2^{{\text{cf}}(\kappa )},\kappa ^{+})$
global choice: The axiom of global choice says there is a well ordering of the class of all sets
Godel
Gödel: 1. Kurt Gödel; 2. A Gödel number is a number assigned to a formula; 3. The Gödel universe is another name for the constructible universe; 4. Gödel's incompleteness theorems show that sufficiently powerful consistent recursively enumerable theories cannot be complete; 5. Gödel's completeness theorem states that consistent first-order theories have models

H

𝔥: The distributivity number
H: Abbreviation for "hereditarily"
H_κ
H(κ): The set of sets that are hereditarily of cardinality less than κ
Hartogs: 1. Friedrich Hartogs; 2. The Hartogs number of a set X is the least ordinal α such that there is no injection from α into X.
Hausdorff: 1. Felix Hausdorff; 2. A Hausdorff gap is a gap in the ordered set of growth rates of sequences of integers, or in a similar ordered set
HC: The set of hereditarily countable sets
hereditarily: If P is a property the a set is hereditarily P if all elements of its transitive closure have property P. Examples: Hereditarily countable set Hereditarily finite set
Hessenberg: 1. Gerhard Hessenberg; 2. The Hessenberg sum and Hessenberg product are commutative operations on ordinals
HF: The set of hereditarily finite sets
Hilbert: 1. David Hilbert; 2. Hilbert's paradox states that a Hotel with an infinite number of rooms can accommodate extra guests even if it is full
HS: The class of hereditarily symmetric sets
HOD: The class of hereditarily ordinal definable sets
huge cardinal: A huge cardinal is a cardinal number κ such that there exists an elementary embedding j : V → M with critical point κ from V into a transitive inner model M containing all sequences of length j(κ) whose elements are in M
hyperarithmetic: A hyperarithmetic set is a subset of the natural numbers given by a transfinite extension of the notion of arithmetic set
hyperinaccessible
hyper-inaccessible: 1. "Hyper-inaccessible cardinal" usually means a 1-inaccessible cardinal; 2. "Hyper-inaccessible cardinal" sometimes means a cardinal κ that is a κ-inaccessible cardinal; 3. "Hyper-inaccessible cardinal" occasionally means a Mahlo cardinal
hyper-Mahlo: A hyper-Mahlo cardinal is a cardinal κ that is a κ-Mahlo cardinal
hyperverse: The hyperverse is the set of countable transitive models of ZFC

I

𝔦: The independence number
I0, I1, I2, I3: The rank-into-rank large cardinal axioms
ideal: An ideal in the sense of ring theory, usually of a Boolean algebra, especially the Boolean algebra of subsets of a set
iff: if and only if
inaccessible cardinal: A (weakly or strongly) inaccessible cardinal is a regular uncountable cardinal that is a (weak or strong) limit
indecomposable ordinal: An indecomposable ordinal is a nonzero ordinal that is not the sum of two smaller ordinals, or equivalently an ordinal of the form ω^α or a gamma number.
independence number: The independence number 𝔦 is the smallest possible cardinality of a maximal independent family of subsets of a countable infinite set
indescribable cardinal: An indescribable cardinal is a type of large cardinal that cannot be described in terms of smaller ordinals using a certain language
individual: Something with no elements, either the empty set or an urelement or atom
indiscernible: A set of indiscernibles is a set I of ordinals such that two increasing finite sequences of elements of I have the same first-order properties
inductive: A poset is called inductive if every non-empty ordered subset has an upper bound
ineffable cardinal: An ineffable cardinal is a type of large cardinal related to the generalized Kurepa hypothesis whose consistency strength lies between that of subtle cardinals and remarkable cardinals
inner model: An inner model is a transitive model of ZF containing all ordinals
Int: Interior of a subset of a topological space
internal: An archaic term for extensional (relation)

J

j: An elementary embedding
J: Levels of the Jensen hierarchy
Jensen: 1. Ronald Jensen; 2. The Jensen hierarchy is a variation of the constructible hierarchy; 3. Jensen's covering theorem states that if 0^# does not exist then every uncountable set of ordinals is contained in a constructible set of the same cardinality
Jónsson: 1. Bjarni Jónsson; 2. A Jónsson cardinal is a large cardinal such that for every function f: [κ]^<ω → κ there is a set H of order type κ such that for each n, f restricted to n-element subsets of H omits at least one value in κ.; 3. A Jónsson function is a function $f:[x]^{\omega }\to x$ with the property that, for any subset y of x with the same cardinality as x, the restriction of $f$ to $[y]^{\omega }$ has image $x$ .

K

Kelley: 1. John L. Kelley; 2. Morse–Kelley set theory, a set theory with classes
KH: Kurepa's hypothesis
kind: Ordinals of the first kind are successor ordinals, and ordinals of the second kind are limit ordinals or 0
KM: Morse–Kelley set theory
Kleene–Brouwer ordering: The Kleene–Brouwer ordering is a total order on the finite sequences of ordinals
KP: Kripke–Platek set theory
Kripke: 1. Saul Kripke; 2. Kripke–Platek set theory consists roughly of the predicative parts of set theory
Kurepa: 1. Đuro Kurepa; 2. The Kurepa hypothesis states that Kurepa trees exist; 3. A Kurepa tree is a tree (T, <) of height $\omega _{1}$ , each of whose levels is countable, with at least $\aleph _{2}$ branches

L

L: 1. L is the constructible universe, and L_α is the hierarchy of constructible sets; 2. L_κλ is an infinitary language
large cardinal: 1. A large cardinal is type of cardinal whose existence cannot be proved in ZFC.; 2. A large large cardinal is a large cardinal that is not compatible with the axiom V=L
Laver: 1. Richard Laver; 2. A Laver function is a function related to supercompact cardinals that takes ordinals to sets
Lebesgue: 1. Henri Lebesgue; 2. Lebesgue measure is a complete translation-invariant measure on the real line
LEM: Law of the excluded middle
Lévy: 1. Azriel Lévy; 2. The Lévy collapse is a way of destroying cardinals; 3. The Lévy hierarchy classifies formulas in terms of the number of alternations of unbounded quantifiers
lightface: The lightface classes are collections of subsets of an effective Polish space definable by second-order formulas without parameters (as opposed to the boldface hierarchy that allows parameters). They include the arithmetical, hyperarithmetical, and analytical sets
limit: 1. A (weak) limit cardinal is a cardinal, usually assumed to be nonzero, that is not the successor κ⁺ of another cardinal κ; 2. A strong limit cardinal is a cardinal, usually assumed to be nonzero, larger than the powerset of any smaller cardinal; 3. A limit ordinal is an ordinal, usually assumed to be nonzero, that is not the successor α+1 of another ordinal α
limited: A limited quantifier is the same as a bounded quantifier
LM: Lebesgue measure
local: A property of a set x is called local if it has the form ∃δ V_δ⊧ φ(x) for some formula φ
LOTS: Linearly ordered topological space
Löwenheim: 1. Leopold Löwenheim; 2. The Löwenheim–Skolem theorem states that if a first-order theory has an infinite model then it has a model of any given infinite cardinality
LST: The language of set theory (with a single binary relation ∈)

M

m: 1. A measure; 2. A natural number
𝔪: The smallest cardinal at which Martin's axiom fails
M: 1. A model of ZF set theory; 2. M_α is an old symbol for the level L_α of the constructible universe
MA: Martin's axiom
MAD: Maximally Almost Disjoint
Mac Lane: 1. Saunders Mac Lane; 2. Mac Lane set theory is Zermelo set theory with the axiom of separation restricted to formulas with bounded quantifiers
Mahlo: 1. Paul Mahlo; 2. A Mahlo cardinal is an inaccessible cardinal such that the set of inaccessible cardinals less than it is stationary
Martin: 1. Donald A. Martin; 2. Martin's axiom for a cardinal κ states that for any partial order P satisfying the countable chain condition and any family D of dense sets in P of cardinality at most κ, there is a filter F on P such that F ∩ d is non-empty for every d in D; 3. Martin's maximum states that if D is a collection of $\aleph _{1}$ dense subsets of a notion of forcing that preserves stationary subsets of ω₁, then there is a D-generic filter
meager
meagre: A meager set is one that is the union of a countable number of nowhere-dense sets. Also called a set of first category.
measure: 1. A measure on a σ-algebra of subsets of a set; 2. A probability measure on the algebra of all subsets of some set; 3. A measure on the algebra of all subsets of a set, taking values 0 and 1
measurable cardinal: A measurable cardinal is a cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. Most (but not all) authors add the condition that it should be uncountable
mice: Plural of mouse
Milner–Rado paradox: The Milner–Rado paradox states that every ordinal number α less than the successor κ⁺ of some cardinal number κ can be written as the union of sets X1,X2,... where Xn is of order type at most κⁿ for n a positive integer.
MK: Morse–Kelley set theory
MM: Martin's maximum
morass: A morass is a tree with ordinals associated to the nodes and some further structure, satisfying some rather complicated axioms.
Morse: 1. Anthony Morse; 2. Morse–Kelley set theory, a set theory with classes
Mostowski: 1. Andrzej Mostowski; 2. The Mostowski collapse is a transitive class associated to a well founded extensional set-like relation.
mouse: A certain kind of structure used in constructing core models; see mouse (set theory)
multiplicative axiom: An old name for the axiom of choice

N

N: 1. The set of natural numbers; 2. The Baire space ω^ω
naive set theory: 1. Naive set theory can mean set theory developed non-rigorously without axioms; 2. Naive set theory can mean the inconsistent theory with the axioms of extensionality and comprehension; 3. Naive set theory is an introductory book on set theory by Halmos
natural: The natural sum and natural product of ordinals are the Hessenberg sum and product
NCF: Near Coherence of Filters
non: non(I) is the uniformity of I, the smallest cardinality of a subset of X not in the ideal I of subsets of X
nonstat
nonstationary: 1. A subset of an ordinal is called nonstationary if it is not stationary, in other words if its complement contains a club set; 2. The nonstationary ideal I_NS is the ideal of nonstationary sets
normal: 1. A normal function is a continuous strictly increasing function from ordinals to ordinals; 2. A normal filter or normal measure on an ordinal is a filter or measure closed under diagonal intersections; 3. The Cantor normal form of an ordinal is its base ω expansion.
NS: Nonstationary
null: German for zero, occasionally used in terms such as "aleph null" (aleph zero) or "null set" (empty set)
number class: The first number class consists of finite ordinals, and the second number class consists of countable ordinals.

O

OCA: The open coloring axiom
OD: The ordinal definable sets
Omega logic: Ω-logic is a form of logic introduced by Hugh Woodin
On: The class of all ordinals
ordinal: 1. An ordinal is the order type of a well-ordered set, usually represented by a von Neumann ordinal, a transitive set well ordered by ∈.; 2. An ordinal definable set is a set that can be defined by a first-order formula with ordinals as parameters
ot: Abbreviation for "order type of"

P

𝔭: The pseudo-intersection number, the smallest cardinality of a family of infinite subsets of ω that has the strong finite intersection property but has no infinite pseudo-intersection.
P: 1. The powerset function; 2. A poset
pairing function: A pairing function is a bijection from X×X to X for some set X
pantachie
pantachy: A pantachy is a maximal chain of a poset
paradox: 1. Berry's paradox; 2. Burali-Forti's paradox; 3. Cantor's paradox; 4. Hilbert's paradox; 5. Milner–Rado paradox; 6. Richard's paradox; 7. Russell's paradox; 8. Skolem's paradox
partial order: 1. A set with a transitive antisymmetric relation; 2. A set with a transitive symmetric relation
partition cardinal: An alternative name for an Erdős cardinal
PCF: Abbreviation for "possible cofinalities", used in PCF theory
PD: The axiom of projective determinacy
perfect set: A perfect set is a subset of a topological set equal to its derived set
permutation model: A permutation model of ZFA is constructed using a group
PFA: The proper forcing axiom
PM: The hypothesis that all projective subsets of the reals are Lebesgue measurable
po: An abbreviation for "partial order" or "poset"
poset: A set with a partial order
Polish space: A Polish space is a separable topological space homeomorphic to a complete metric space
pow: Abbreviation for "power (set)"
power: "Power" is an archaic term for cardinality
power set
powerset: The powerset or power set of a set is the set of all its subsets
projective: 1. A projective set is a set that can be obtained from an analytic set by repeatedly taking complements and projections; 2. Projective determinacy is an axiom asserting that projective sets are determined
proper: 1. A proper class is a class that is not a set; 2. A proper subset of a set X is a subset not equal to X.; 3. A proper forcing is a forcing notion that does not collapse any stationary set; 4. The proper forcing axiom asserts that if P is proper and D_α is a dense subset of P for each α<ω₁, then there is a filter G $\subseteq$ P such that D_α ∩ G is nonempty for all α<ω₁
PSP: Perfect subset property

Q

Q: The (ordered set of) rational numbers
QPD: Quasi-projective determinacy
quantifier: ∀ or ∃
Quasi-projective determinacy: All sets of reals in L(R) are determined

R

𝔯: The unsplitting number
R: 1. R_α is an alternative name for the level V_α of the von Neumann hierarchy.; 2. The set of real numbers, usually stylized as $\mathbb {R}$
Ramsey: 1. Frank P. Ramsey; 2. A Ramsey cardinal is a large cardinal satisfying a certain partition condition
ran: The range of a function
rank: 1. The rank of a set is the smallest ordinal greater than the ranks of its elements; 2. A rank V_α is the collection of all sets of rank less than α, for an ordinal α; 3. rank-into-rank is a type of large cardinal (axiom)
reflecting cardinal: A reflecting cardinal is a type of large cardinal whose strength lies between being weakly compact and Mahlo
reflection principle: A reflection principle states that there is a set similar in some way to the universe of all sets
regressive: A function f from a subset of an ordinal to the ordinal is called regressive if f(α)<α for all α in its domain
regular: A regular cardinal is one equal to its own cofinality.
Reinhardt cardinal: A Reinhardt cardinal is a cardinal in a model V of ZF that is the critical point of an elementary embedding of V into itself
relation: A set or class whose elements are ordered pairs
Richard: 1. Jules Richard; 2. Richard's paradox considers the real number whose nth binary digit is the opposite of the nth digit of the nth definable real number
RO: The regular open sets of a topological space or poset
Rowbottom: 1. Frederick Rowbottom; 2. A Rowbottom cardinal is a large cardinal satisfying a certain partition condition
rud: The rudimentary closure of a set
rudimentary: A rudimentary function is a functions definable by certain elementary operations, used in the construction of the Jensen hierarchy
Russell: 1. Bertrand Russell; 2. Russell's paradox is that the set of all sets not containing themselves is contradictory so cannot exist

S

𝔰: The splitting number
SBH: Stationary basis hypothesis
SCH: Singular cardinal hypothesis
SCS: Semi-constructive system
Scott: 1. Dana Scott; 2. Scott's trick is a way of coding proper equivalence classes by sets by taking the elements of the class of smallest rank
second: 1. A set of second category is a set that is not of first category: in other words a set that is not the union of a countable number of nowhere-dense sets.; 2. An ordinal of the second class is a countable infinite ordinal; 3. An ordinal of the second kind is a limit ordinal or 0; 4. Second order logic allows quantification over subsets as well as over elements of a model
sentence: A formula with no bound variables
separating set: 1. A separating set is a set containing a given set and disjoint from another given set; 2. A separating set is a set S of functions on a set such that for any two distinct points there is a function in S with different values on them.
separative: A separative poset is one that can be densely embedded into the poset of nonzero elements of a Boolean algebra.
set: A collection of distinct objects, considered as an object in its own right.
SFIP: Strong finite intersection property
SH: Suslin's hypothesis
Shelah: 1. Saharon Shelah; 2. A Shelah cardinal is a large cardinal that is the critical point of an elementary embedding satisfying certain conditions
shrewd cardinal: A shrewd cardinal is a type of large cardinal generalizing indecribable cardinals to transfinite levels
Sierpinski
Sierpiński: 1. Wacław Sierpiński; 2. A Sierpiński set is an uncountable subset of a real vector space whose intersection with every measure-zero set is countable
Silver: 1. Jack Silver; 2. The Silver indiscernibles form a class I of ordinals such that I∩L_κ is a set of indiscernibles for L_κ for every uncountable cardinal κ
singular: 1. A singular cardinal is one that is not regular; 2. The singular cardinal hypothesis states that if κ is any singular strong limit cardinal, then 2^κ = κ⁺.
SIS: Semi-intuitionistic system
Skolem: 1. Thoralf Skolem; 2. Skolem's paradox states that if ZFC is consistent there are countable models of it; 3. A Skolem function is a function whose value is something with a given property if anything with that property exists; 4. The Skolem hull of a model is its closure under Skolem functions
small: A small large cardinal axiom is a large cardinal axiom consistent with the axiom V=L
SOCA: Semi open coloring axiom
Solovay: 1. Robert M. Solovay; 2. The Solovay model is a model of ZF in which every set of reals is measurable
special: A special Aronszajn tree is one with an order preserving map to the rationals
square: The square principle is a combinatorial principle holding in the constructible universe and some other inner models
standard model: A model of set theory where the relation ∈ is the same as the usual one.
stationary set: A stationary set is a subset of an ordinal intersecting every club set
strong: 1. The strong finite intersection property says that the intersection of any finite number of elements of a set is infinite; 2. A strong cardinal is a cardinal κ such that if λ is any ordinal, there is an elementary embedding with critical point κ from the universe into a transitive inner model containing all elements of V_λ; 3. A strong limit cardinal is a (usually nonzero) cardinal that is larger than the powerset of any smaller cardinal
strongly: 1. A strongly inaccessible cardinal is a regular strong limit cardinal; 2. A strongly Mahlo cardinal is a strongly inaccessible cardinal such that the set of strongly inaccessible cardinals below it is stationary; 3. A strongly compact cardinal is a cardinal κ such that every κ-complete filter can be extended to a κ complete ultrafilter
subtle cardinal: A subtle cardinal is a type of large cardinal closely related to ethereal cardinals
successor: 1. A successor cardinal is the smallest cardinal larger than some given cardinal; 2. A successor ordinal is the smallest ordinal larger than some given ordinal
such that: A condition used in the definition of a mathematical object
sunflower: A sunflower, also called a delta system, is a collection of sets such that any two distinct sets have intersection X for some fixed set X
Souslin
Suslin: 1. Mikhail Yakovlevich Suslin (sometimes written Souslin); 2. A Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition; 3. A Suslin cardinal is a cardinal λ such that there exists a set P ⊂ 2^ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ.; 4. The Suslin hypothesis says that Suslin lines do not exist; 5. A Suslin line is a complete dense unbounded totally ordered set satisfying the countable chain condition; 6. The Suslin number is the supremum of the cardinalities of families of disjoint open non-empty sets; 7. The Suslin operation, usually denoted by A, is an operation that constructs a set from a Suslin scheme; 8. The Suslin problem asks whether Suslin lines exist; 9. The Suslin property states that there is no uncountable family of pairwise disjoint non-empty open subsets; 10. A Suslin representation of a set of reals is a tree whose projection is that set of reals; 11. A Suslin scheme is a function with domain the finite sequences of positive integers; 12. A Suslin set is a set that is the image of a tree under a certain projection; 13. A Suslin space is the image of a Polish space under a continuous mapping; 14. A Suslin subset is a subset that is the image of a tree under a certain projection; 15. The Suslin theorem about analytic sets states that a set that is analytic and coanalytic is Borel; 16. A Suslin tree is a tree of height ω₁ such that every branch and every antichain is at most countable.
supercompact: A supercompact cardinal is an uncountable cardinal κ such that for every A such that Card(A) ≥ κ there exists a normal measure over [A]^κ.
super transitive
supertransitive: A supertransitive set is a transitive set that contains all subsets of all its elements
symmetric model: A symmetric model is a model of ZF (without the axiom of choice) constructed using a group action on a forcing poset

T

𝔱: The tower number
T: A tree
tall cardinal: A tall cardinal is a type of large cardinal that is the critical point of a certain sort of elementary embedding
Tarski: 1. Alfred Tarski; 2. Tarski's theorem states that the axiom of choice is equivalent to the existence of a bijection from X to X×X for all sets X
TC: The transitive closure of a set
total order: A total order is a relation that is transitive and antisymmetric such that any two elements are comparable
totally indescribable: A totally indescribable cardinal is a cardinal that is Π^m
_n-indescribable for all m,n
transfinite: 1. An infinite ordinal; 2. Transfinite induction is induction over ordinals
transitive: 1. A transitive relation; 2. The transitive closure of a set is the smallest transitive set containing it.; 3. A transitive set or class is a set or class such that the membership relation is transitive on it.; 4. A transitive model is a model of set theory that is transitive and has the usual membership relation
tree: 1. A tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <; 2. A tree is a collection of finite sequences such that every prefix of a sequence in the collection also belongs to the collection.; 3. A cardinal κ has the tree property if there are no κ-Aronszajn trees
type class: A type class or class of types is the class of all order types of a given cardinality, up to order-equivalence.

U

𝔲: The ultrafilter number, the minimum possible cardinality of an ultrafilter base
Ulam: 1. Stanislaw Ulam; 2. An Ulam matrix is a collection of subsets of a cardinal indexed by pairs of ordinals, that satisfies certain properties.
Ult: An ultrapower or ultraproduct
ultrafilter: 1. A maximal filter; 2. The ultrafilter number 𝔲 is the minimum possible cardinality of an ultrafilter base
ultrapower: An ultraproduct in which all factors are equal
ultraproduct: An ultraproduct is the quotient of a product of models by a certain equivalence relation
unfoldable cardinal: An unfoldable cardinal a cardinal κ such that for every ordinal λ and every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j(κ) ≥ λ.
uniformity: The uniformity non(I) of I is the smallest cardinality of a subset of X not in the ideal I of subsets of X
uniformization: Uniformization is a weak form of the axiom of choice, giving cross sections for special subsets of a product of two Polish spaces
universal
universe: 1. The universal class, or universe, is the class of all sets.; A universal quantifier is the quantifier "for all", usually written ∀
urelement: An urelement is something that is not a set but allowed to be an element of a set

V

V: V is the universe of all sets, and the sets V_α form the Von Neumann hierarchy
V=L: The axiom of constructibility
Veblen: 1. Oswald Veblen; 2. The Veblen hierarchy is a family of ordinal valued functions, special cases of which are called Veblen functions.
von Neumann: 1. John von Neumann; 2. A von Neumann ordinal is an ordinal encoded as the union of all smaller (von Neumann) ordinals; 3. The von Neumann hierarchy is a cumulative hierarchy V_α with V_α+1 the powerset of V_α.
Vopenka
Vopěnka: 1. Petr Vopěnka; 2. Vopěnka's principle states that for every proper class of binary relations there is one elementarily embeddable into another; 3. A Vopěnka cardinal is an inaccessible cardinal κ such that and Vopěnka's principle holds for V_κ

W

weakly: 1. A weakly inaccessible cardinal is a regular weak limit cardinal; 2. A weakly compact cardinal is a cardinal κ (usually also assumed to be inaccessible) such that the infinitary language L_κ,κ satisfies the weak compactness theorem; 3. A weakly Mahlo cardinal is a cardinal κ that is weakly inaccessible and such that the set of weakly inaccessible cardinals less than κ is stationary in κ
well founded: A relation is called well founded if every non-empty subset has a minimal element
well ordering: A well ordering is a well founded relation, usually also assumed to be a total order
Wf: The class of well-founded sets, which is the same as the class of all sets if one assumes the axiom of foundation
Woodin: 1. Hugh Woodin; 2. A Woodin cardinal is a type of large cardinal that is the critical point of a certain sort of elementary embedding, closely related to the axiom of projective determinacy

XYZ

Z: Zermelo set theory without the axiom of choice
ZC: Zermelo set theory with the axiom of choice
Zermelo: 1. Ernst Zermelo; 2. Zermelo−Fraenkel set theory is the standard system of axioms for set theory; 3. Zermelo set theory is similar to the usual Zermelo-Fraenkel set theory, but without the axioms of replacement and foundation; 4. Zermelo's well-ordering theorem states that every set can be well ordered
ZF: Zermelo−Fraenkel set theory without the axiom of choice
ZFA: Zermelo−Fraenkel set theory with atoms
ZFC: Zermelo−Fraenkel set theory with the axiom of choice
ZF-P: Zermelo−Fraenkel set theory without the axiom of choice or the powerset axiom
Zorn: 1. Max Zorn; 2. Zorn's lemma states that if every chain of a non-empty poset has an upper bound then the poset has a maximal element

References

Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.

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Glossary of set theory

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