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. (shrink)

D’Aquino et al. (J Symb Log 75(1):1–11, 2010) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by IΔ0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss’ bounded arithmetic: PV or $${\Sigma^b_1-IND^{|x|_k}}$$. It also holds for IΔ0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the other hand, it (...) fails for the extension of IOpen by an axiom expressing the Bézout property, even under the same assumption on cofinality. (shrink)

The question of whether the bounded arithmetic theories and are equal is closely connected to the complexity question of whether is equal to. In this paper, we examine the still open question of whether the prenex version of,, is equal to. We give new dependent choice‐based axiomatizations of the ‐consequences of and. Our dependent choice axiomatizations give new normal forms for the ‐consequences of and. We use these axiomatizations to give an alternative proof of the finite axiomatizability of and to (...) show new results such as is finitely axiomatized and that there is a finitely axiomatized theory,, containing and contained in. On the other hand, we show that our theory for splits into a natural infinite hierarchy of theories. We give a diagonalization result that stems from our attempts to separate the hierarchy for. (shrink)