  The parameter-free part $\text{PA}_2^\ast$ of $\text{PA}_2$, the 2nd order
Peano arithmetic, is considered. We make use of a product/iterated Sacks
forcing to define an $\omega$-model of $\text{PA}_2^\ast +
\text{CA}(\Sigma^1_2)$, in which an example of the full Comprehension schema
$\text{CA}$ fails. Using Cohen's forcing, we also define an $\omega$-model of
$\text{PA}_2^\ast$, in which not every set has its complement, and hence the
full $\text{CA}$ fails in a rather elementary way.
