Axiom of the empty set

The axiom of the empty set is, in Mathématiques one of the possible Axiome S of the axiomatic Théorie of the units. As its name indicates it, it makes it possible to pose the existence of a Empty set. In the modern presentations, it is not mentioned any more among the axioms of the set theories of Zermelo, or Zermelo-Fraenkel, because it is consequence in Logique first order Schéma of axioms of comprehension.

Exposure

In the formal Language of the axioms of Zermelo-Frankel, the axiom is written:

$\ exists has \ \ forall B (B \ not \ in A)$

or in other words:

There exists a unit has such as, for any unit B , B is not an element of has , i.e. there exists a unit whose no whole is element.

The Axiome of extensionnality can be used to show that this unit is single. It is called the Empty set and it is noted $\ empty$ or {}.

Primarily, the axiom thus affirms that the empty set exists.

Empty set and diagram of comprehension

The existence of the empty set can be shown by comprehension, and thus does not have to belong to the axioms of the set theory of Zermelo or Zermelo-Fraenkel, when those are seen like first order theories. Indeed, in Logical first order, the fields of interpretation of the variables of basic objects, here of the overall variables, is not vacuums. That would complicate much the talk of the logical rules to consider empty fields. It is what allows the introduction of new variables into the reasoning: as soon as a new variable is introduced, it is supposed that it indicates an object.

It is thus enough, in the case which worries us, to apply the Schéma of axioms of comprehension to an arbitrary unit, for a property ever carried out: either there a unit, has = { X ∈ there | X ≠ X} is well the empty set, i.e. $\ forall X (X \ not \ in a)$ .

See too

axiomatic Set theory

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