# Proof

(related to Theorem: Trichotomy of Ordinals)

• Let $$\alpha, \beta$$ be fixed ordinals and $$\gamma=\alpha\cap \beta$$.
• By the lemma about equivalence of set inclusion and element inclusion of ordinals, it follows from $$\gamma\subseteq\alpha$$ that $$\gamma\in\alpha$$ or $$\gamma=\alpha$$, and from $$\gamma\subseteq\beta$$ that $$\gamma\in\beta$$ or $$\gamma=\beta$$.
• Altogether, we have the following cases:
• $$\gamma\in\alpha$$ and $$\gamma=\beta$$. From this case it follows that $$\beta\in\alpha$$.
• $$\gamma\in\beta$$ and $$\gamma=\alpha$$. From this case it follows that $$\alpha\in\beta$$.
• $$\gamma=\beta$$ and $$\gamma=\alpha$$. From this case it follows that $$\alpha=\beta$$, and by the axiom of foundation, neither $$\alpha\in \beta$$ nor $$\beta\in \alpha$$ is possible.
• $$\gamma\in\beta$$ and $$\gamma\in\alpha$$. However, this case is not possible, since otherwise we would have $$\gamma\in\alpha\cap\beta=\gamma$$, contradicting the axiom of foundation.

Github: ### 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