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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 $z\cup \{z\}.$

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

| | | | | created: 2014-06-09 21:52:37 | modified: 2019-08-03 19:22:43 | by: bookofproofs | references: [656], [983]