Conservative fragments of ${{S}^{1}_{2}}$ and ${{R}^{1}_{2}}$

Abstract

Conservative subtheories of ${{R}^{1}_{2}}$ and ${{S}^{1}_{2}}$ are presented. For ${{S}^{1}_{2}}$, a slight tightening of Jeřábek’s result (Math Logic Q 52(6):613–624, 2006) that ${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$ is presented: It is shown that ${T^{0}_{2}}$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this ${\forall\Sigma^{b}_{1}}$-theory, we define a ${\forall\Sigma^{b}_{0}}$-theory, ${T^{-1}_{2}}$, for the ${\forall\Sigma^{b}_{0}}$-consequences of ${S^{1}_{2}}$. We show ${T^{-1}_{2}}$ is weak by showing it cannot ${\Sigma^{b}_{0}}$-define division by 3. We then consider what would be the analogous ${\forall\hat\Sigma^{b}_{1}}$-conservative subtheory of ${R^{1}_{2}}$ based on Pollett (Ann Pure Appl Logic 100:189–245, 1999. It is shown that this theory, ${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$, also cannot ${\Sigma^{b}_{0}}$-define division by 3. On the other hand, we show that ${{S}^{0}_{2}+open_{\{||id||\}}}$-COMP is a ${\forall\hat\Sigma^{b}_{1}}$-conservative subtheory of ${R^{1}_{2}}$. Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium’ 98, 262–279, 2000) and show that ${\hat{C}^{0}_{2}}$ is ${\forall\hat\Sigma^{b}_{1}}$-conservative over a theory based on open _cl-comprehension.