Transitive Class/Examples/Ordinal 3 without 1 is not Transitive

Example of Non-Transitive Class

Let $\O$ denote the empty class.

Consider the class $S$, defined as:

$S := \set {\O, \set {\O, \set \O} }$

$S$ is not transitive.

Proof

$S$ has $2$ elements: $\O$ and $\set {\O, \set \O}$.

Note that one of the elements of $\set {\O, \set \O}$ is $\set \O$.

But $\set \O$ 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