Axiom of extensionality

The extensionality is an axiom of set theory, which was formulated in 1888 by Richard Dedekind and states that two classes or quantities are equal if and only if they have the same elements. From Dedekind Ernst Zermelo took over the extensionality in the first axiomatic set theory, the Zermelo set theory of 1907., From where it came into the extended Zermelo -Fraenkel set theory ZF and all later versions of axiomatic set theory.

Clarification

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

In quantity gauges with primal elements, the variables are restricted to quantities about in ZFUs:

In amount Teaching with classes that extensionality is used more generally with free class variables, such as the Ackermann set theory:

Importance

The extensionality guarantees the uniqueness of a class or set whose elements is described by a property of its elements, ie, by a condition of the form

With the extensionality and the usual abstraction principle it follows the equality:

This uniqueness results in particular for the empty set axiom pair axiom, power set axiom, Union axiom, axiom, Axiom replacement required quantities and there allowed the introduction of the usual class notations.