Some sources mention instead of the axiom of existence that postulate the existence of a set, the following axiom that postulates the existence of the empty set. Both axioms are equivalent, i.e. can be exchanged. This is because the existence of the empty set implies the existence of any set. On the other hand, the axiom of separation can be used to separate the empty set from any other given set.

Axiom: Axiom of Empty Set

There is an (improper) set, called the null set or the empty set, denoted by \(\emptyset\), which does not contain any elements.

\[\exists X~\forall z~z\notin X.\]


Corollaries: 1

Axioms: 1 2
Corollaries: 3
Definitions: 4 5
Proofs: 6 7 8

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  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  2. Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition