Greatest Element/Examples/Finite Subsets of Natural Numbers

Examples of Greatest Elements

Let $\FF$ denote the set of finite subsets of the natural numbers $\N$.

Consider the ordered set $\struct {\FF, \subseteq}$.

$\struct {\FF, \subseteq}$ has no greatest element.

Proof

Aiming for a contradiction, suppose $A \in \FF$ is the greatest element of $\struct {\FF, \subseteq}$.

From Greatest Element is Maximal, $A$ is a maximal element of $\struct {\FF, \subseteq}$.

But from Maximal Element: Finite Subsets of Natural Numbers, $\struct {\FF, \subseteq}$ has no maximal element.

Hence $A$ cannot be the greatest element of $\struct {\FF, \subseteq}$.

The result follows from Proof by Contradiction.

$\blacksquare$

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $6 \ \text {(a)}$