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...

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

\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...

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...

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...

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:...

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...

whose members are precisely the members of A that satisfy φ {\displaystyle \varphi } . By the axiom of extensionality this set is unique. We usually denote...

, 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 )...

be equal if they have all the same members. This is called the axiom of extensionality. In English, the word equal is derived from the Latin aequālis...

(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...

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...

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...

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 ≡...

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

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...

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...

Axiom of choice Axiom of constructibility Axiom of extensionality Axiom of infinity Axiom of limitation of size Axiom of pairing Axiom of union Axiom...

of the elements of A {\displaystyle A} . Then one can use the axiom of extensionality to show that this set is unique. For readability, define the binary...

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

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...

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...

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...

an axiom is a premise or starting point for reasoning. In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom". Logical axioms are...

capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory. This kind of extension is used so constantly...

mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty...

satisfy the axiom of extensionality. A set further is hereditarily ordinal definable if it is ordinal definable and all elements of its transitive closure...

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