Glossary of set theory
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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 Γ_{0} is the Feferman–Schütte ordinal.
- δ
- 1. A delta number is an ordinal of the form ω^{ωα}
- 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 ^{ω}ω (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. 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 I_{1} 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
- 1. Cardinal exponentiation
- 2. Ordinal exponentiation
- 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
- The cardinality of a set X
- 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
- The Hebrew letter aleph, which indexes the aleph numbers or infinite cardinals ℵ_{α}
- The Hebrew letter beth, which indexes the beth numbers ב_{α}
- A serif form of the Hebrew letter gimel, representing the gimel function
- 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 ℵ_{0}.
- 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".^{ [1]}
- 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 x^{2}=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^{ωα}
- 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 is a sequence of subsets of an ordinal , then the diagonal intersection is
- diamond principle
- Jensen's diamond principle states that there are sets A_{α}⊆α for α<ω_{1} such that for any subset A of ω_{1} the set of α with A∩α = A_{α} is stationary in ω_{1}.
- 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 (ε_{0}) 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 Γ_{0} 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^{ℵα} = 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
- 2. The gimel function
- 3. The gimel hypothesis states that
- 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 with the property that, for any subset y of x with the same cardinality as x, the restriction of to has image .
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 , each of whose levels is countable, with at least 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 dense subsets of a notion of forcing that preserves stationary subsets of ω_{1}, 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 κ^{n} 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<ω_{1}, then there is a filter G P such that D_{α} ∩ G is nonempty for all α<ω_{1}
- 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
- 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
- rudimentary set theory
- See basic set theory.
- 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 ω_{1} 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
See also
References
- ^ P. Aczel, The Type Theoretic Interpretation of Constructive Set Theory (1978)
- 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.