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...

have only countably many formulas, every notion of definable numbers has at most countably many definable real numbers. However, by Cantor's diagonal argument...

up definable in Wiktionary, the free dictionary. In mathematical logic, the word definable may refer to: A definable real number A definable set A definable...

{\displaystyle \mathbb {R} } to define a curve. In general, definable sets without quantifiers are easy to describe, while definable sets involving possibly nested...

sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets...

complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics. In set theory as...

organizational change and performance Ordinal definable set, a set requiring only finitely-many ordinals to define under first-order logic. OD (video game)...

In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every...

projective sets of reals; however for reasons related to Tarski's undefinability theorem the notion of a definable set of reals cannot be defined in the language...

portal Mathematics portal Analytic proposition Circular definition Definable set Definitionism Denotation Extensional definition Fallacies of definition...

tunnel information before the IPv4 client address. Thus there is no definable set of prefixes more specific than 2001:0::/32 for Teredo packets with Martian...

arithmetical property. A set Y of natural numbers is implicitly arithmetical or implicitly arithmetically definable if it is definable with an arithmetical...

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...

and R {\displaystyle R} that defines a set of integers that is not definable using only addition. Presburger-definable relations admit another characterization:...

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...

\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...

Set (/sɛt/; Egyptological: Sutekh - swtẖ ~ stẖ or: Seth /sɛθ/) is a god of deserts, storms, disorder, violence, and foreigners in ancient Egyptian religion...

which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar...

definable by language. Curry's paradox List of self–referential paradoxes Kleene–Rosser paradox List of paradoxes Löb's theorem Ordinal definable set...

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...

This is also known as set comprehension and set abstraction. Set-builder notation can be used to describe a set that is defined by a predicate, that is...

equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence...

{M}}} is said to be definable (or explicitly definable cf. Beth definability, or ∅ {\displaystyle \emptyset } -definable, or definable with parameters from...

definable, it is enough that B be a set; the axiom of specification essentially says that A must then also be a set. As with subsets, the empty set is...

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...

of only two sets. Ice Age and Alliances were the first two sets to have a well-defined relationship, but the idea of calling connected sets a "block" or...

Skolemization Lindenbaum–Tarski algebra Löb's theorem Arithmetical set Definable set Ehrenfeucht–Fraïssé game Herbrand interpretation / Herbrand structure...

the set of all natural numbers. Other recursively defined mathematical objects include factorials, functions (e.g., recurrence relations), sets (e.g...

called A-definable if it is contained in Sn(A); in that case A is called a set of parameters for D. A subset is called definable if it is A-definable for some...