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Proof

(related to Lemma: Successor of Ordinal)

• According to the a property of transitive sets, stating that with any transitive set \(X\) also \(X\cup\{X\}\) must be transitive.
• Thus, the building rule \(s(\alpha)=\alpha\cup\{\alpha\}\) ensures the transitivity of \(s(\alpha)\).
• Since all of its elements $x\in s(\alpha)$ are also transitive sets, it must be an ordinal, because of equivalent notions of ordinals.
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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. Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018