\begin{frame}
  \frametitle{Ordinals}

  \begin{definition}
  A set $S$ is \alert{transitive} if $x \in S$ implies $x \subseteq S$.
  \pause\medskip
  
  An \alert{ordinal} is a transitive set whose elements are transitive sets.
  \end{definition}
  \pause

  \begin{example}[$0,1,2,3,\ldots$]
  $\varnothing$,\; 
  $\{\varnothing\}$,\; 
  $\{\varnothing,\{\varnothing\}\}$,\; 
  $\{\varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\}\} \}$, \ldots
  \end{example}
  \pause\medskip
  
  \begin{definition}
  We define $\alpha < \beta \;\Longleftrightarrow\; \alpha \in \beta$. 
  \end{definition}
  \pause
  
  \begin{lemma}
  The relation $<$ is a total order on ordinals.
  \end{lemma}
  \pause
  
  \begin{lemma}
  For every ordinal $\beta$, we have $\beta = \{\alpha \mid \alpha < \beta\}$.
  \end{lemma}
\end{frame}