The parameterfree Comprehension does not imply the full Comprehension in the 2nd order Peano arithmetic
Vladimir Grigor'evich Kanovei, Vassily Lyubetsky · arXiv (Cornell University) · 2022
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 $ω$-model of $\text{PA}_2^\ast + \text{CA}(Σ^1_2)$, in which an example of the full Comprehension schema $\text{CA}$ fails. Using Cohen's forcing, we also define an $ω$-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.