All of the axioms introduce so far do not ensure the existence of infinite sets. The following axiom closes this gap.

Axiom: Axiom of Infinity

There exists a set \(X\) containing the empty set and also with every element $z$ also the element $z\cup \{z\}.$

\[\exists X~(\emptyset \in X \wedge \forall~z(z\in X \Rightarrow z\cup \{z\}\in X).\]


Corollaries: 1

  1. Definition: Inductive Set
  2. Definition: Minimal Inductive Set

Axioms: 1
Definitions: 2 3 4 5
Examples: 6

Thank you to the contributors under CC BY-SA 4.0!




  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