Transitive Class/Examples/Singleton of Singleton of Empty Class is not Transitive
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Example of Non-Transitive Class
Let $\O$ denote the empty set.
Consider the class $S$, defined as:
- $S := \set {\set \O}$
$S$ is not transitive.
Proof
$S$ has $1$ element: $\set \O$.
$\set \O$ in turn has one element: $\O$.
But $\O$ is not itself an element of $S$.
Thus not all elements of elements of $S$ are themselves elements of $S$.
Hence by definition $S$ is not transitive.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 10$ Some useful facts about transitivity: Exercise $10.1 \ \text {(a)}$