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While the existence of inductive sets is ensured by the axiom of infinity, there is one particular inductive set worth a closer look - the minimal inductive set. We can use the axiom of separation, to define it uniquely:

Definition: Minimal Inductive Set

The set $\omega:=\{W\mid \forall X(X\text{ is an inductive set }\Rightarrow W\in X)\}$ is the minimal set, which fulfills the axiom of infinity. It is called the minimal inductive set.

Mentioned in:

Corollaries: 1
Definitions: 2 3
Examples: 4
Explanations: 5
Proofs: 6

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References

Bibliography

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