H. Ki and T. Linton showed that for any integer $b \geq 2$,
the set of normal numbers in base $b$ are $\Pi^0_3$-complete. Other
authors have since used this result to find the complexity of other
sets of normal numbers. Let $(X,T,\mu)$ be a dynamical system with $X$
a Polish space, $T$ continuous, and $\mu$ a shift invariant
probability measure. We generalize Ki and Linton's result by showing
that if $(X,T)$ has more than one invariant measure and satisfies a
weak form of the specification property then the set of $\mu$-generic
points is $\Pi^0_3$-complete. As a consequence we establish that the
sets of normal numbers with respect to a wide class of expansions
other than the $b$-ary expansion, for example the continued fraction
expansion or the $\beta$-expansion for irrational $\beta$, are also
$\Pi^0_3$-complete