\begin{frame}
  \frametitle{Examples}

  \begin{alertblock}{}
  Give a counter-model for:
  \begin{talign}
    &\myall{x}{\myex{y}{ R(x,y)}}, \\ 
    &\myall{x}{\myall{y}{\myall{z}{\big(R(x,y) \wedge R(y,z) \to R(x,z)\big)}}} \\
    &\quad \models\;\; \myex{x}{R(x,x)}
  \end{talign}
  \end{alertblock}  
  \pause\bigskip
   
  What do the premises it mean?
  \begin{enumerate}[(a)]
  \pause
    \item Every object has a successor.
  \pause
    \item The successor-relation is transitive. \\\pause
          Hence any $n$-step successor is an immediate successor.
  \end{enumerate}
  \pause
  What does the conclusion mean?
  \begin{itemize}
  \pause
    \item There is an object that is its own successor.
  \end{itemize}
  \pause\bigskip
  
  \begin{alertblock}{}
    Can there be finite counter-models?
  \end{alertblock}
  \pause
  
  No, because by (a) there would by cycles, and by (b) every 
  element on a cycle would be its own successor.
  \vspace{10cm}
\end{frame}