Collection of All Ordered Sets is not Set
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Theorem
Let $\mathrm {OS}$ denote the collection of all ordered sets.
Then $\mathrm {OS}$ is not a set.
Proof
Let $C$ be the collection of all singletons:
- $\set {x: \exists y: x = \set y}$
Define a mapping $\map f {\set y} = \RR$ where $\RR$ is a reflexive relation on $\set y$.
By Reflexive Relation on Singleton is Well-Ordering, $\RR$ is an ordering.
Thus:
- $f: C \to \mathrm {OS}$
- $\map f {y_1} = \map f {y_2} \implies y_1 = y_2$
Thus $f$ is an injection.
By definition of cardinality, $C$ is the collection of all sets which are equivalent to $\set \O$.
By Collection of Sets Equivalent to Set Containing Empty Set is Proper Class, $C$ is proper.
Thus, by Injection from Proper Class to Class, $\mathrm {OS}$ is proper.
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 $25$