Axiom of extensionality

The axiom of extensionality is an axiom of set theory that was formulated by Richard Dedekind in 1888 and says that two classes or sets are equal if and only if they have the same elements. Ernst Zermelo adopted the axiom of extensionality from Dedekind in the first axiomatic set theory , the Zermelo set theory of 1907. From there it came into the expanded Zermelo-Fraenkel set theory ZF and all later versions of the axiomatic set theory.

Clarification

In the predicate logic form of the Zermelo-Fraenkel set theory ZF, which is authoritative today , in which all objects are sets, the axiom of extensionality is formally:

${\displaystyle \forall A,B\colon (A=B\iff \forall C\colon (C\in A\iff C\in B))}$

In set theory with primitive elements , the variables are restricted to sets, for example in ZFU :

${\displaystyle A{\text{ ist Menge }}\land B{\text{ ist Menge }}\Rightarrow (A=B\iff \forall C\colon (C\in A\iff C\in B))}$

In set theory with classes , the axiom of extensionality is used more generally with free class variables, for example in Ackermann set theory or in class logic :

${\displaystyle A=B\iff \forall C\colon (C\in A\iff C\in B)}$

meaning

The axiom of extensionality guarantees the uniqueness of a class or set , the elements of which are described by a property of its elements , i.e. by a condition of the form ${\displaystyle M}$ ${\displaystyle A(x)}$

${\displaystyle \forall x\colon (x\in M\iff A(x))}$

With the axiom of extensionality and the usual abstraction principle, equality then follows:

${\displaystyle M\,=\,\{x\mid A(x)\}}$

This ambiguity arises in particular for the empty set axiom , pair amount Axiom , power set axiom , union axiom , axiom , axiom schema of replacement required quantities and there allowed the introduction of the ordinary class spelling.

Individual evidence

1. ↑ Richard Dedekind : What are and what are the numbers? Vieweg, Braunschweig 1888, § 1.2, quote: "The system S is therefore the same as the system T , in sign S = T , if every element of S is also an element of T and every element of T is also an element of S. " online .
2. ↑ Ernst Zermelo : Investigations on the basics of set theory. (1907). In: Mathematical Annals . Vol. 65, 1908, pp. 261–281, there Axiom II p. 263, the axiom of determinateness , of which Dedekind speaks. Zermelo mentions Dedekind as a role model.