  We study the question of when a given countable ordinal $\alpha$ is
$\Sigma^1_n$- or $\Pi^1_n$-reflecting in models which are neither $\mathsf{PD}$
models nor the constructible universe, focusing on generic extensions of $L$.
We prove, amongst other things, that adding any number of Cohen or random
reals, or forcing with Sacks forcing or any lightface Borel weakly homogeneous
ccc forcing notion cannot change such reflection properties. Moreover we show
that collapse forcing increases the value of the least reflecting ordinals but,
curiously, to ordinals which are still smaller than the $\omega_1$ of $L$.
