This is a glossary of terms and definitions related to the topic 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 ωα.
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 ωωα. 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 ωε=ε.
η 1. The order type of the rational numbers. 2. An eta set, a type of ordered set. 3. ηα 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) → (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. ωα is an alternative name for ℵα, used when it is considered as an ordinal number rather than a cardinal number.
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 first-order set theoretical symbols with their usual definitions, (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 definitions; and (conjunction), or (inclusive disjunction), implies (conditional), is equivalent to (biconditional), not (negation), for all (universal quantifier), there exists (existential quantifier).
≡ An equivalence, or congruence relation.
↾ f↾X once denoted the corestriction of a relation, or mapping, but in modern mathematics is the restriction of a relation, or mapping f, to some set X, as-in, 1.) f↾X≔{(x,y):x∈X and y∈X} — the restriction, and corestriction — for a relation f to X. 2.) f↾X≔{f(x):x∈X} — the restriction of the domain — for a mapping f to X.
↿ f↿X is the restriction of a relation or mapping f, to some set X.
△ (A triangle, not to be confused with the Greek letter Δ) 1. For X△Y, the symmetric difference of sets X, and Y, as-in, X△Y≔{x:x∈X-Y or x∈Y-X}. 2. A diagonal intersection.
◊ The diamond principle.
♣ A clubsuit principle.
□ The square principle, or Q.E.D. — quod erat demonstrandum — that which was to be demonstrated.
∘ The composition of functions
⁀ 1. α⁀b is the extension of a sequence a, by an element b, as-in, 2. For an α-sequence, a, a⁀b≔a⨿{b} such that b is the α-th element of a.
+ 1. Addition of ordinals. 2. Addition of cardinals. 3. α+ is the successor cardinal, or the least 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. For sets a, and b, a~b is the set of elements of a not in b, called the difference of, a and b, as-in, 2. a~b≔{α:α∈a and α∉b}. 3. An equivalence relation.
∖ Synonym of ~.
− Synonym of ~.
≈ Has the same cardinality as.
× For sets a, and b, a×b≔{(c,d):c∈a and d∈b}, called the product of sets a, and b.
/ For a set a, and an equivalence relation ~, a/~≔{b/~:b∈a}, called the quotient of a set by an equivalence relation ~.
× For ordinals a, and b, a×b≔𝙾𝚛𝚍(a×b) called the ordinal product of ordinals a, and b.
⊗ For cardinals a, and b, a⊗b≔𝙲𝚛𝚍(a×b), called the cardinal product of cardinals a, and b.
* 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
α β {\displaystyle \alpha ^{\beta }}
1. Cardinal exponentiation 2. Ordinal exponentiation
β α {\displaystyle {}^{\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)nm 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 | {\displaystyle |X|}
The cardinality of a set X
‖ φ ‖ {\displaystyle \|\varphi \|}
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 {\displaystyle \aleph } The Hebrew letter aleph, which indexes the aleph numbers or infinite cardinals ℵα
ℶ {\displaystyle \beth } The Hebrew letter beth, which indexes the beth numbers בα
ℷ {\displaystyle \gimel }
A serif form of the Hebrew letter gimel, representing the gimel function ℷ ( κ ) = κ cf κ {\displaystyle \gimel (\kappa )=\kappa ^{\operatorname {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. ACω 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 Lα is an admissible set
AH The generalized continuum hypothesis states that 2ℵα = ℵα+1
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
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 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. Sometimes also called "rudimentary set theory".
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
beth 1. The Hebrew letter ב 2. A beth number בα
Beth Evert Willem Beth, as in Beth definability
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
Cartesian product The set of all ordered pairs obtained from two sets, where each pair consists of one element from each 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)
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 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
class comprehension schema A principle in set theory allowing the formation of classes based on properties or conditions that their members satisfy.
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 cofinality
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.
cofinite Referring to a set whose complement in a larger set is finite, often used in discussions of topology and set theory.
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
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 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 Lα of the constructible hierarchy is isomorphic to an element Lγ 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
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 A core model is a special sort of inner model generalizing the constructible universe
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 A term used for the countable chain condition by authors who think terminology should be logical
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 The countable chain condition (ccc) for a poset states that every antichain is countable
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, ω {\displaystyle \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 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
Dedekind 1. Richard Dedekind 2. A Dedekind-infinite set is a set that can be put into a one-to-one correspondence with one of its proper subsets, indicating a type of infinity; a Dedekind-finite set is a set that is not Dedekind-infinite. (These are also spelled without the hyphen, as "Dedekind finite" and "Dedekind infinite".)
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 ωωα 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
dependent choice See Axiom of dependent choice
determinateness See Axiom of extensionality
Df The set of definable subsets of a set
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 If ⟨ X α ∣ α < δ ⟩ {\displaystyle \displaystyle \langle X_{\alpha }\mid \alpha <\delta \rangle }
is a sequence of subsets of an ordinal δ {\displaystyle \displaystyle \delta } , then the diagonal intersection Δ α < δ X α , {\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha },}
is
{ β < δ ∣ β ∈ ⋂ α < β X α } . {\displaystyle \displaystyle \{\beta <\delta \mid \beta \in \bigcap _{\alpha <\beta }X_{\alpha }\}.}
diamond principle Jensen's diamond principle states that there are sets Aα⊆α for α<ω1 such that for any subset A of ω1 the set of α with A∩α = Aα is stationary in ω1.
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 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"
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 An elementary embedding is a function preserving all properties describable in the language of set theory
empty set The unique set that contains no elements, denoted by ∅ {\displaystyle \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 1. An epsilon number is an ordinal α such that α=ωα 2. Epsilon zero (ε0) is the smallest epsilon number
equicardinal Synonym of equinumerous
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
equipotent 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 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
Euler diagram 1. A graphical representation of the logical relationships between sets, using overlapping circles to illustrate intersections, unions, and complements of sets.
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 Vκ+η into some Vλ 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 Fσ 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 ∀∃∧∨¬
foundation axiom See Axiom of foundation
Fraenkel Abraham Fraenkel
G
𝖌 The groupwise density number
G 1. A generic ultrafilter 2. A Gδ is a countable intersection of open sets
gamma number A gamma number is an ordinal of the form ωα
GCH Generalized continuum hypothesis
generalized continuum hypothesis The generalized continuum hypothesis states that 2אα = אα+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 ℷ {\displaystyle \gimel }
2. The gimel function ℷ ( κ ) = κ cf ( κ ) {\displaystyle \gimel (\kappa )=\kappa ^{{\text{cf}}(\kappa )}}
3. The gimel hypothesis states that ℷ ( κ ) = max ( 2 cf ( κ ) , κ + ) {\displaystyle \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
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 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"
Hκ 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 1. 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 2. An ω-huge cardinal is a large cardinal related to the I1 rank-into-rank axiom
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
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 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
improper See proper, below.
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 ωα 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 1. An inductive set is a set that can be generated from a base set by repeatedly applying a certain operation, such as the set of natural numbers generated from the number 0 by the successor operation. 2. An inductive definition is a definition that specifies how to construct members of a set based on members already known to be in the set, often used for defining recursively defined sequences, functions, and structures. 3. A poset is called inductive if every non-empty ordered subset has an upper bound
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 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
integers The set of whole numbers including positive, negative, and zero, denoted by ℤ.
internal An archaic term for extensional (relation)
intersection The set containing all elements that are members of two or more sets, denoted by A ∩ B {\displaystyle 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 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
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 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 ] ω → x {\displaystyle 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 ] ω {\displaystyle [y]^{\omega }} has image x.
K
Kelley 1. John L. Kelley 2. Morse–Kelley set theory (also called Kelley–Morse 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
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–Platek set theory
Kripke 1. Saul Kripke 2. Kripke–Platek set theory consists roughly of the predicative parts of set theory
Kuratowski 1. Kazimierz Kuratowski 2. A Kuratowski ordered pair is a definition of an ordered pair using only set theoretical concepts, specifically, the ordered pair (a, b) is defined as the set {{a}, {a, b}}. 3. "Kuratowski-Zorn lemma" is an alternative name for Zorn's lemma
Kurepa 1. Đuro Kurepa 2. The Kurepa hypothesis states that Kurepa trees exist 3. A Kurepa tree is a tree (T, <) of height ω 1 {\displaystyle \omega _{1}} , each of whose levels is countable, with at least ℵ 2 {\displaystyle \aleph _{2}} branches
L
L 1. L is the constructible universe, and Lα is the hierarchy of constructible sets 2. Lκλ 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
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 1. Richard Laver 2. A Laver function is a function related to supercompact cardinals that takes ordinals to sets
least upper bound The smallest element in a pa