  A Cantor series expansion for a real number $x$ with respect to a basic
sequence $Q=(q_1,q_2,\dots)$, where $q_i \geq 2$, is a representation of the
form $x=a_0 + \sum_{i=1}^\infty \frac{a_i}{q_1q_2\cdots q_i}$ where $0 \leq
a_i<q_i$. These generalize ordinary base $b$ expansions where $q_i=b$. Ki and
Linton showed that for ordinary base $b$ expansions the set of normal numbers
is a $\boldsymbol{\Pi}^0_3$-complete set, establishing the exact complexity of
this set. In the case of Cantor series there are three natural notions of
normality: normality, ratio normality,and distribution normality (these notions
are equivalent for base $b$ expansions). We show that for any $Q$ the set
$\mathscr{DN}(Q)$ of distribution normal number is
$\boldsymbol{\Pi}^0_3$-complete, and if $Q$ is $1$-divergent (i.e.,
$\sum_{i=1}^\infty \frac{1}{q_i}$ diverges) then the sets $\mathscr{N}(Q)$ and
$\mathscr{RN}(Q)$ of normal and ratio normal numbers are
$\boldsymbol{\Pi}^0_3$-complete. We further show that all five non-trivial
differences of these sets are $D_2(\boldsymbol{\Pi}^0_3)$-complete if $\lim_i
q_i=\infty$ and $Q$ is $1$-divergent (the trivial case is
$\mathscr{N}(Q)\setminus \mathscr{RN}(Q)=\emptyset$). This shows that except
for the containment $\mathscr{N}(Q)\subseteq \mathscr{RN}(Q)$, these three
notions are as independent as possible.
