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

The parameter-free part PA_2^* of PA_2 , second order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an ω -model of PA_2^*+ CA(Σ ^1_2) , in which an example of the full Comprehension schema CA fails. Using Cohen’s forcing, we also define an ω -model of PA_2^* , in which not every set has its complement, and hence the full CA fails in a rather elementary way.