In mathematical logic, a definable set is an n-ary relation on the domain of a structure whose elements satisfy some formula in the first-order language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining the relation.
variables in the formula. A set A {\displaystyle A} is definable in M {\displaystyle {\mathcal {M}}} without parameters if it is definable in M {\displaystyle...
extensionality. A set further is hereditarily ordinal definable if it is ordinal definable and all elements of its transitive closure are ordinal definable. The class...
up definable in Wiktionary, the free dictionary. In mathematical logic, the word definable may refer to: A definable real number A definableset A definable...
have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument...
{\displaystyle \mathbb {R} } to define a curve. In general, definablesets without quantifiers are easy to describe, while definablesets involving possibly nested...
well-ordered set, usually represented by a von Neumann ordinal, a transitive set well ordered by ∈. 2. An ordinal definableset is a set that can be defined by...
arithmetical property. A set Y of natural numbers is implicitly arithmetical or implicitly arithmetically definable if it is definable with an arithmetical...
and R {\displaystyle R} that defines a set of integers that is not definable using only addition. Presburger-definable relations admit another characterization:...
cases: The empty set is computable. The entire set of natural numbers is computable. Each natural number (as defined in standard set theory) is computable;...
themselves (Russell), of the set of all transfinite ordinal numbers (Burali-Forti), and the set of all finitely definable real numbers (Richard)." He goes...
projective sets of reals; however for reasons related to Tarski's undefinability theorem the notion of a definableset of reals cannot be defined in the language...
organizational change and performance Ordinal definableset, a set requiring only finitely-many ordinals to define under first-order logic. OD (video game)...
the set of all natural numbers. Other recursively defined mathematical objects include factorials, functions (e.g., recurrence relations), sets (e.g...
image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF. The axiom schema is motivated...
equivalence relation on a setdefines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence...
closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and...
which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar...
mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set along with a distance defined between any two points)...
sequences definable relative to M and that set may not even exist in some such M. Similarly, the map from the set of formulas that define integer sequences...
tunnel information before the IPv4 client address. Thus there is no definableset of prefixes more specific than 2001:0::/32 for Teredo packets with Martian...
made up of similar atoms, but their structures do not follow an easily definableset of rules. The isostere concept was formulated by Irving Langmuir in...
hierarchy if it is definable by a formula of second-order arithmetic with only existential set quantifiers and no other set quantifiers. A set is classified...
\Pi _{n}^{0}} formulas that define the set. For example, the set of odd natural numbers n {\displaystyle n} is definable by either ∀ k ( n ≠ 2 × k ) {\displaystyle...
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