Transitive set - Example According to my notes, a set $A$ is called transitive if the elements of its elements are elements of $A$.
For example,  the set of natural numbers $\omega$ is a transitive set.
Also, if $n \in \omega$ then $n$ is a transitive set since $n=\{0,1,2, \dots, n-1 \}$ and if we take a $k \in n$ then $k=\{0,1,2, \dots, k-1 \}$.
Could you give me an example of an other transitive set? Or can a transitive set only contain natural numbers?
 A: If $A$ is a transitive set, then $\mathcal P(A)$ is a transitive set. Take now $\omega$ which is a transitive set, then $\mathcal P(\omega)$ is a transitive set, but its elements are sets of natural numbers, rather than just natural numbers.
A: The ordinal numbers are exactly the transitive and $\in$-well-ordered sets. So every ordinal (including the natural numbers) is transitive. However, there is a strong claim in the reverse direction as well:

For any set $x$, there is a transitive set that contains $x$.

In particular, since the intersection of transitive sets is transitive, the intersection of all such sets is a well-defined object called the transitive closure $TC(x)$, and it is always transitive and contains $x$. Thus, you can pick any non-ordinal and take its transitive closure, and this set will be transitive and contain non-ordinals. The smallest non-ordinal set is $\{1\}=\{\{\emptyset\}\}$, so $TC(\{1\})$, which can be explicitly constructed by just starting with $\{\{1\}\}$ and taking elements of elements (of elements...) and adding them to the set, produces a non-ordinal transitive set.
$$TC(\{1\})=\{\{1\},1,0\}=\{\{\{\emptyset\}\},\{\emptyset\},\emptyset\}$$
Here's a more complicated example:
\begin{align}
TC(\{1,2,\{2\}\})&=\{\{1,2,\{2\}\}\}\cup\{1,2,\{2\}\}\cup\{0,1,2\}\cup\{0,1\}\cup\{0\}\cup\emptyset\\
&=\{\{1,2,\{2\}\},\{2\},2,1,0\}
\end{align}
