Maximal Element under Subset Relation need not be Greatest Element
Jump to navigation
Jump to search
Theorem
Let $A$ be a class.
Let $M \in A$ be a maximal element of $A$ under the subset relation
Then $M$ is not necessarily the greatest element of $A$.
Proof
Let $A = \set {x, y}$ such that:
- $x = \set \O$
- $y = \set {\set \O}$
Then:
- $x$ and $y$ are both maximal elements of $A$ by definition.
However:
- $x \not \subseteq y$
and:
- $y \not \subseteq x$
and so neither $x$ nor $y$ are the greatest element of $A$.
$\blacksquare$
Also see
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles