Glossary of set theory

This is a glossary of set theory.

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$α$: Often used for an ordinal
$β$: 1. $β$ $X$ 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 $ω ω α$; 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 $ω ω$ (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 $κ (α) \to (α) < ω$
$λ$: 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 $I 1$ 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

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∈, =, ⊆, ⊇, ⊃, ⊂, ∪, ∩, ∅: 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)
$\equiv$: 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 $κ \to(λ) n m$ 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
$\|\varphi \|$: 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
$\aleph$: The Hebrew letter aleph, which indexes the aleph numbers or infinite cardinals $ℵ α$
$\beth$: The Hebrew letter beth, which indexes the beth numbers $ב α$
$\gimel$: A serif form of the Hebrew letter gimel, representing the gimel function $\gimel (\kappa )=\kappa ^{\operatorname {cf} \kappa }$
$ת$: The Hebrew letter Taw, used by Cantor for the class of all cardinal numbers

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