The language $\{ww \mid w \in \{0,1\}^{*} \}$ is not a CFL
You're assuming that, when you expand $S\to S_1S_1$, the two copies of $S_1$ must be expanded in the same way, to give two copies of some string&nbsp;$w$. But it doesn't have to be that way.  For example, you could do
$$S\to S_1S_1 \to 0S_1S_1 \to 0 S_1 \to 01S_1\to 01\,.$$