Conservative fragments of S12 and R12

Conservative subtheories of R 2 and S 1 2 are presented. For S 1 2 , a slight tightening of Jeřábek’s result [18] that T 0 2 ∀Σ1 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 ∀Σ1-theory, we define a ∀Σ0-theory, T−1 2 , for the ∀Σ0-consequences of S 1 2 . We show T −1 2 is weak by showing it cannot Σ b 0-define division by 3. We then consider what would be the analogous ∀Σ̂1-conservative subtheory of R 2 based on Pollett [24]. It is shown that this theory, T 0,{2 ̇ (||id||)} 2 , also cannot Σ b 0-define division by 3. On the other hand, we show that S 0 2 +open{||id||}-COMP is a ∀Σ̂1-conservative subtheory of R 2 . Finally, we give a refinement of Johannsen and Pollett [21] and show that Ĉ 0 2 is ∀Σ̂1-conservative over a theory based on openclcomprehension. Mathematics Subject Classification: 03F30, 68Q15