Transitive Class/Examples

Examples of Transitive Classes

Empty Class

The empty class is transitive.

Singleton of Empty Class

Let $\O$ denote the empty class.

Then the singleton $\set \O$ is transitive.

Class $\set {\O, \set \O}$

Let $\O$ denote the empty class.

Then the class:

$\set {\O, \set \O}$

is transitive.

Class $\set {\O, \set \O, \set {\O, \set \O} }$

Let $\O$ denote the empty class.

Consider the ordinal $3$, defined as:

$\mathcal 3 := \set {\O, \set \O, \set {\O, \set \O} }$

$\mathcal 3$ is transitive.

Class $\set {\O, \set {\O, \set \O} }$

Let $\O$ denote the empty class.

Consider the class $S$, defined as:

$S := \set {\O, \set {\O, \set \O} }$

$S$ is not transitive.

Singleton of Singleton of Empty Class is not Transitive

Let $\O$ denote the empty set.

Consider the class $S$, defined as:

$S := \set {\set \O}$

$S$ is not transitive.