Abstract

Lyubetsky and Kanovei showed in [8] that there is a second-order arithmetic model of $\mathrm {Z}_2^{-p}$, (comprehension for all second-order formulas without parameters), in which $\Sigma ^1_2$ - $\mathrm {CA}$ (comprehension for all $\Sigma ^1_2$ -formulas with parameters) holds, but $\Sigma ^1_4$ - $\mathrm {CA}$ fails. They asked whether there is a model of $\mathrm {Z}_2^{-p}+\Sigma ^1_2$ - $\mathrm {CA}$ with the optimal failure of $\Sigma ^1_3$ - $\mathrm {CA}$. We answer the question positively by constructing such a model in a forcing extension by a tree iteration of Jensen’s forcing. Let $\mathrm {Coll}^{-p}$ be the parameter-free collection scheme for second-order formulas and let $\mathrm {AC}^{-p}$ be the parameter-free choice scheme. We show that there is a model of $\mathrm {Z}_2^{-p}+\mathrm { AC}^{-p}+\Sigma ^1_2$ - $\mathrm {CA}$ with a failure of $\Sigma ^1_4$ - $\mathrm {CA}$. We also show that there is a model of $\mathrm {Z}_2^{-p}+\mathrm {Coll}^{-p}+\Sigma ^1_2$ - $\mathrm {CA}$ with a failure of $\Sigma ^1_4$ - $\mathrm {CA}$ and a failure of $\mathrm {AC}^{-p}$, so that, in particular, the schemes $\mathrm {Coll}^{-p}$ and $\mathrm {AC}^{-p}$ are not equivalent over $\mathrm {Z}_2^{-p}$.