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Lemma: Comparing the Elements of Strictly Ordered Sets

In a strictly ordered set $(V,\prec)$, for all elements \(a,b\in V\) exactly one of the following cases holds:

• Either $a \prec b$ ($a$ is smaller than $b$),
• or $a \succ b$ ($a$ is greater than $b$).
• else $a=b$ ($a$ equals $b$).

Because of these three possibilities, the strict order "$\prec$" is sometimes also called a trichotomous order.

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References

Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition
2. Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018