It was shown by H. Ki and T. Linton that for any integer $b \geq 2$, the set of numbers that are normal in base $b$ is a $\Pi_3^0$-complete set.  Other authors have since used this result to arrive at the complexity of other sets of normal numbers.  Let $(X,T,\mu)$ be a dynamical system, that is,  $X$ is a Polish space, $T\colon X\to X$ is continuous, and $\mu$ is an invariant Borel probability measure on $X$. We generalize the result of H. Ki and T. Linton by showing that the set of generic points of $(X,T,\mu)$ is a $\Pi_3^0$-complete set if $(X,T)$ satisfies a weakening of the specification property and there are at least two invariant measures. We also prove a stronger result for systems which are subshifts of shift spaces on a finite or countable alphabet. We extend the result from the case of a single measure $\mu$ to a closed, connected, proper set $V$ of measures. Here the corresponding set $G(V)$ is also $\Pi_3^0$-hard, but to show $G(V)$ is a $\Pi_3^0$ set requires $X$ being compact. We give an example of a non-compact $(X,T)$ for which $G(V)$ is $\Pi_1^1$-complete. As a consequence of our results we answer a question of Sharkovsky and Sivak.