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: 2018-05-05 23:29:10 | by: *bookofproofs* | references: [656], [983]

## 1.**Definition**: Inductive Set

## 2.**Definition**: Minimal Inductive Set

## 3.**Corollary**: Minimal Inductive Set Is Subset Of All Inductive Sets

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[983] **Ebbinghaus, H.-D.**: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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