We have defined the set $\omega$ as minimal with respect to the inclusion "$\in$" in relation to all inductive sets. The following corollary states that $\omega$ is also minimal, if we look at that relation of being the subset $\subseteq$ of inductive sets:# Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets

(related to Axiom: Axiom of Infinity)

The minimal inductive set $\omega$ is a subset of all inductive sets, formally: $$\forall W(W\text{ is inductive }\Rightarrow \omega\subseteq W).$$

Proofs: 1

Explanations: 1


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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.: "Einf├╝hrung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition