Abstract

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