The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory...

\lnot (u\in u)\}.} Thus, the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique...

Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity...

, the power set of x {\displaystyle x} , consisting precisely of the subsets of x {\displaystyle x} . By the axiom of extensionality, the set P ( x )...

implies both propositional and functional extensionality. Extensionality principles are usually assumed as axioms, especially in type theories where computational...

axiom of extensionality to show that this set C is unique. We call the set C the pair of A and B, and denote it {A,B}. Thus the essence of the axiom is:...

B)\to A=B.} This axiom is identical to the axiom of extensionality found in many other set theories, including ZF. Any element or a subset of a set is a set...

assume any axioms except the axiom of extensionality and the axiom of induction—a natural number is either zero or a successor and each of its elements...

{\displaystyle S} of A whose members are precisely the members of A that satisfy φ ( x ) {\displaystyle \varphi (x)} . By the axiom of extensionality this set...

the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does. The type theoretical...

Look up extension, extend, or extended in Wiktionary, the free dictionary. Extension, extend or extended may refer to: Axiom of extensionality Extensible...

(See the Lévy hierarchy.) Axiom of extensionality: Two sets are the same if and only if they have the same elements. Axiom of induction: φ(a) being a formula...

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty...

results from the axiom of extensionality that is often expressed as "two sets that have the same elements are equal".) The truth of an equality depends...

words: There is a set such that no element is a member of it. We can use the axiom of extensionality to show that there is only one empty set. Since it is...

existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty...

x_{n})].} Then the axiom schema of replacement is replaced by a single axiom that uses a class. Finally, ZFC's axiom of extensionality is modified to handle...

interpreted in a weak set theory whose axioms are extensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34). In fact...

axiom schema of replacement is derivable in S+ + Extensionality. Hence S+ + Extensionality has the power of ZF. Boolos also argued that the axiom of choice...

z\in y.} Identical to Extensionality above. I would be identical to the axiom of extensionality in ZFC, except that the scope of I includes proper classes...

that is, if every element of A is an element of B and every element of B is an element of A. (See axiom of extensionality.) Thus a set is completely...

predicate. AXIOM I. Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa ... then M ≡...

form. In particular the equivalence holds in the presence of the axioms of extensionality, pairing, union and powerset. ∀ A ( [ ∀ x ∃ ! y ϕ ( x , y ...

ontology as ZFC). Axiom of extensionality: Two sets are identical if they have the same members. Axiom of regularity: No set is a member of itself, and circular...

the axiom of extensionality must be formulated to apply only to objects that are not urelements. This situation is analogous to the treatments of theories...

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written...

are using the same element relation and no new sets were added. Axiom of extensionality: Two sets are the same if they have the same elements. If x {\displaystyle...

fail as badly as it can (or rather, as extensionality permits): Boffa's axiom implies that every extensional set-like relation is isomorphic to the elementhood...

non-logical predicate ∈ {\displaystyle \in } , and that includes the axiom of extensionality: ∀ x ∀ y ( ∀ z ( z ∈ x ⟺ z ∈ y ) ⟹ x = y ) {\displaystyle \forall...

an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. An axiom schema is a formula in the metalanguage of an axiomatic...