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$n-1$-elementig ist und $x_{0}\in X_{0}$ und $G(x_{0})$, + gilt per IV (\textdagger)~$\forall{x\in X_{0}:~}G(x)$. Jetzt betrachten wir die rechte Teilmenge, $X_{1}$.\\ \fbox{Wähle nun einen Fisch, $\tilde{x}\in X_{0}$ mit $\tilde{x}\neq x_{0}$.}\\ Wegen (\textdagger) gilt $G(\tilde{x})$.\\ Da $\tilde{x}\neq x_{0}$, liegt dieser Fisch nun in der Auswahlmenge $X_{1}$.\\ Also ist $X_{1}$ eine $n-1$-elementige Menge und $x_{1}\in X_{1}$ und $G(x_{1})$.\\ - Per Induktionsvoraussetzung gilt also (\ddag)~$\forall{x\in X_{1}:~}G(x)$. + Per IV gilt also $\forall{x\in X_{1}:~}G(x)$.\\ + Daraus folgt $\forall{x\in X:~}G(x)$, da $X=X_{1}\cup\{x_{0}\}$. - Aus (\textdagger) und (\ddag) folgt - $\forall{x\in X:~}G(x)$, da ja $X=X_{0}\cup X_{1}$. - - Also gilt $\Phi(n)$. + Darum gilt $\Phi(n)$. \end{kompaktenum} Darum gilt $\forall{n\in\ntrlpos:~}\Phi(n)$. \end{proof} - Das Problem mit diesem Argument steckt in dem Induktionsschritt beim Schritt: + Das Problem mit diesem Argument steckt im Induktionsschritt an genau dieser Stelle: \begin{quote} Wähle nun einen Fisch, $\tilde{x}\in X_{0}$ mit $\tilde{x}\neq x_{0}$.