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For example it is well known that in univalent foundations, the univalence axiom implies both propositional and functional extensionality. Extensionality principles are usually assumed as axioms, especially in type theories where computational content must be preserved. However, in set theory and other extensional foundations, functional ...
The axiom of extensionality, [1] [2] also called the axiom of extent, [3] [4] is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. [5] [6] The axiom defines what a set is. [1] Informally, the axiom means that the two sets A and B are equal if and only if A and B have the same members.
An extensional definition gives meaning to a term by specifying its extension, that is, every object that falls under the definition of the term in question.. For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class.
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction.
For example, it is common in naive set theory to introduce a symbol for the set that has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant ∅ {\displaystyle \emptyset } and the new axiom ∀ x ( x ∉ ∅ ) {\displaystyle \forall x(x\notin \emptyset )} , meaning "for all x , x is ...
In mathematics, the axiom of power set [1] is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set x {\displaystyle x} the existence of a set P ( x ) {\displaystyle {\mathcal {P}}(x)} , the power set of x {\displaystyle x} , consisting precisely of the subsets of x {\displaystyle x} .
The term extensionality, as used in 'Axiom of Extensionality' has its roots in logic. An intensional definition describes the necessary and sufficient conditions for a term to apply to an object. For example: "An even number is an integer which is divisible by 2." An extensional definition instead lists all objects where the term applies.
Axiom of extensionality; Extensible cardinal; Extension (model theory) Extension (proof theory) Extension (predicate logic), the set of tuples of values that satisfy the predicate; Extension (semantics), the set of things to which a property applies; Extension by definitions
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