Transitive Class/Examples/Ordinal 3

Example of Transitive Class

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.

Proof

Consider the $3$ elements of $\mathcal 3$ in turn.

There are no elements of $\O$.

There is exactly one elements of $\set \O$, and that is $\O$.

We see that $\O$ is itself an element of $\set {\O, \set \O, \set {\O, \set \O} }$.

The elements of $\set {\O, \set \O}$, are $\O$ and $\set \O$.

We see that both are element of $\mathcal 3$.

Thus all elements of elements of $\mathcal 3$ are themselves elements of $\mathcal 3$.

Hence by definition $\mathcal 3$ is 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