Independence of the Comprehension Schema in Second-Order Arithmetic from the Parameter-Free Countable Choice
Vladimir Grigor'evich Kanovei, Vassily Lyubetsky · Mathematical Notes · 2025
We prove that the full Comprehension schema $$\mathbf{CA}$$ in second-order arithmetic $$\mathbf{PA}_2$$ is not provable in the subtheory $$\mathbf{PA}_2^\ast$$ with the parameter-free Comprehension even when adding the parameter-free Countable Choice $$\mathbf{AC}_\omega^\ast$$ and the Comprehension $$\mathbf{CA}(\mathbf\Sigma^1_2)$$ for all $$\mathbf\Sigma^1_2$$ formulas with parameters.