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While the existence of inductive sets is ensured by the axiom of infinity”:https://www.bookofproofs.org/branches/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.

| | | | | created: 2019-01-18 20:04:37 | modified: 2019-01-18 20:10:26 | by: bookofproofs | references: [656], [983]

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Bibliography (further reading)

[983] Ebbinghaus, H.-D.: “Einf├╝hrung 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