\begin{frame}{More Undecidable Problems}
  \begin{block}{}
    \emph{Validity} of a formula $\phi$ in \emph{predicate logic} is \alert{undecidable}.
  \end{block}
%   \ \hfill {\it (Logic and Modelling)}
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  In 1900 \emph{David Hilbert} (1862-1941) formulated 23 scientific problems.
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  Among them the following:
  
  \begin{goal}{}
    \emph{Diophantine equations} consist of polynomials with one or more variables 
    and coefficients in $\mathbb{Z}$. For example:
    \begin{talign}
      3x^2y-7y^2z^3-18 &= 0 \\
      -7y^2+8z^3 &= 0
    \end{talign}
  \end{goal}
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  \begin{block}{}
    \emph{Hilbert's 10th problem}:
    Give an algorithm to decide whether a system of Diophantine equations has a solution in $\mathbb{Z}$.
  \end{block}  
  \pause
  
  In 1970 \emph{Yuri Matiyasevich} proved that this is \alert{undecidable}.
\end{frame}