The parameterfree Comprehension does not imply the full Comprehension in the 2nd order Peano arithmetic

Abstract

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 ${\text{PA}}_2^{\ast }+\text{CA}({\Sigma }_2^1)$, 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.

13 pages