Parameterfree Comprehension Does Not Imply Full Comprehension in Second Order Peano Arithmetic

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