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\begin{document}
\title{Exercices sur les ordinaux}
\author{David A. Madore}
\maketitle

\centerline{\textbf{MITRO206}}

{\footnotesize
\immediate\write18{sh ./vc > vcline.tex}
\begin{center}
Git: \input{vcline.tex}
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\immediate\write18{echo ' (stale)' >> vcline.tex}
\par}

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\exercice

\newcommand{\spaceout}{\hskip1emplus2emminus.5em}

Ranger les ordinaux suivants par ordre croissant :
\spaceout $\omega^{\omega+1} + \omega^\omega\cdot 33$ ;
\spaceout $\omega\cdot 3 + 42$ ;
\spaceout $\omega^{\omega+1} + \omega + 33$ ;
\spaceout $\omega^{\omega+2} + \omega^\omega$ ;
\spaceout $\omega^2\cdot 42 + 1000$ ;
\spaceout $\omega^2 + \omega$ ;
\spaceout $\omega^2\cdot 42 + \omega$ ;
\spaceout $\omega^{\omega^2 + 1}$ ;
\spaceout $\omega^{\omega^{(\omega\cdot 2)}}$ ;
\spaceout $\omega^{\omega^\omega} + 1$ ;
\spaceout $\omega^{\omega+1} + \omega\cdot 33$ ;
\spaceout $\omega^{\omega^2}$ ;
\spaceout $\omega^{\omega^2 + 1} + \omega^{\omega\cdot 2}\cdot 1000$ ;
\spaceout $\omega^{\omega^2 + \omega}$ ;
\spaceout $\omega\cdot 3$ ;
\spaceout $\omega^{(\omega^\omega\cdot 2)}$ ;
\spaceout $\omega^{\omega^3}$ ;
\spaceout $\omega^{\omega+1} + 1000$ ;
\spaceout $\omega^{\omega+2}$ ;
\spaceout $\omega^{\omega+1}\cdot 2$ ;
\spaceout $\omega\cdot 2 + 1729$ ;
\spaceout $\omega^2 + 1000$ ;
\spaceout $42$ ;
\spaceout $\omega^{\omega^2 + \omega} + \omega^{\omega^2 + 1}$ ;
\spaceout $\omega^{\omega\cdot 2}\cdot 1000$ ;
\spaceout $\omega^2\cdot 42$ ;
\spaceout $\omega^{\omega+1} + \omega^2\cdot 33$ ;
\spaceout $\omega^2$ ;
\spaceout $\omega$ ;
\spaceout $\omega^{\omega+1}$ ;
\spaceout $\omega^{\omega^2 + 1} + \omega^{\omega^2}\cdot 42$ ;
\spaceout $\omega^{\omega\cdot 2} + \omega^{\omega+2}$ ;
\spaceout $\omega^{\omega^2 + \omega} + \omega^{\omega^2} + \omega^{\omega+1}$ ;
\spaceout $\omega^{\omega^{(\omega^2)}}$ ;
\spaceout $\omega^{\omega^2 + 1}\cdot 2$ ;
\spaceout $\omega^2 + \omega\cdot 42$ ;
\spaceout $\omega + 42$ ;
\spaceout $\omega^{\omega^2\cdot 2}$ ;
\spaceout $\omega^{\omega\cdot 2 + 42}$ ;
\spaceout $\omega^{\omega\cdot 2}$ ;
\spaceout $\omega\cdot 2$ ;
\spaceout $\omega^{\omega+1} + \omega^2 + 33$ ;
\spaceout $\omega^{\omega^{(\omega+1)}}$ ;
\spaceout $\omega^\omega$ ;
\spaceout $\omega^{\omega^2 + \omega + 1}$ ;
\spaceout $\omega^{\omega^\omega}\cdot 2$ ;
\spaceout $\omega^{\omega^\omega}$ ;
\spaceout $0$ ;
\spaceout $\omega^{\omega^2} + \omega^{\omega+1}$ ;
\spaceout $\omega^{(\omega^\omega + 1)}$.

\begin{corrige}
On vérifie que tous ces ordinaux sont écrits en forme normale de
Cantor (et les exposants de $\omega$ aussi, etc.).  On les compare
donc en comparant à chaque fois la plus grande puissance de $\omega$.

Dans l'ordre croissant : \spaceout $0$ ;
\spaceout $42$ ;
\spaceout $\omega$ ;
\spaceout $\omega + 42$ ;
\spaceout $\omega\cdot 2$ ;
\spaceout $\omega\cdot 2 + 1729$ ;
\spaceout $\omega\cdot 3$ ;
\spaceout $\omega\cdot 3 + 42$ ;
\spaceout $\omega^2$ ;
\spaceout $\omega^2 + 1000$ ;
\spaceout $\omega^2 + \omega$ ;
\spaceout $\omega^2 + \omega\cdot 42$ ;
\spaceout $\omega^2\cdot 42$ ;
\spaceout $\omega^2\cdot 42 + 1000$ ;
\spaceout $\omega^2\cdot 42 + \omega$ ;
\spaceout $\omega^\omega$ ;
\spaceout $\omega^{\omega+1}$ ;
\spaceout $\omega^{\omega+1} + 1000$ ;
\spaceout $\omega^{\omega+1} + \omega + 33$ ;
\spaceout $\omega^{\omega+1} + \omega\cdot 33$ ;
\spaceout $\omega^{\omega+1} + \omega^2 + 33$ ;
\spaceout $\omega^{\omega+1} + \omega^2\cdot 33$ ;
\spaceout $\omega^{\omega+1} + \omega^\omega\cdot 33$ ;
\spaceout $\omega^{\omega+1}\cdot 2$ ;
\spaceout $\omega^{\omega+2}$ ;
\spaceout $\omega^{\omega+2} + \omega^\omega$ ;
\spaceout $\omega^{\omega\cdot 2}$ ;
\spaceout $\omega^{\omega\cdot 2} + \omega^{\omega+2}$ ;
\spaceout $\omega^{\omega\cdot 2}\cdot 1000$ ;
\spaceout $\omega^{\omega\cdot 2 + 42}$ ;
\spaceout $\omega^{\omega^2}$ ;
\spaceout $\omega^{\omega^2} + \omega^{\omega+1}$ ;
\spaceout $\omega^{\omega^2 + 1}$ ;
\spaceout $\omega^{\omega^2 + 1} + \omega^{\omega\cdot 2}\cdot 1000$ ;
\spaceout $\omega^{\omega^2 + 1} + \omega^{\omega^2}\cdot 42$ ;
\spaceout $\omega^{\omega^2 + 1}\cdot 2$ ;
\spaceout $\omega^{\omega^2 + \omega}$ ;
\spaceout $\omega^{\omega^2 + \omega} + \omega^{\omega^2} + \omega^{\omega+1}$ ;
\spaceout $\omega^{\omega^2 + \omega} + \omega^{\omega^2 + 1}$ ;
\spaceout $\omega^{\omega^2 + \omega + 1}$ ;
\spaceout $\omega^{\omega^2\cdot 2}$ ;
\spaceout $\omega^{\omega^3}$ ;
\spaceout $\omega^{\omega^\omega}$ ;
\spaceout $\omega^{\omega^\omega} + 1$ ;
\spaceout $\omega^{\omega^\omega}\cdot 2$ ;
\spaceout $\omega^{(\omega^\omega + 1)}$ ;
\spaceout $\omega^{(\omega^\omega\cdot 2)}$ ;
\spaceout $\omega^{\omega^{(\omega+1)}}$ ;
\spaceout $\omega^{\omega^{(\omega\cdot 2)}}$ ; et enfin
\spaceout $\omega^{\omega^{(\omega^2)}}$.
\end{corrige}


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