On the finite axiomatizability of

Abstract

The question of whether the bounded arithmetic theories S21 and R21 are equal is closely connected to the complexity question of whether boldP is equal to boldNC. In this paper, we examine the still open question of whether the prenex version of R21, sans-serifR̂21, is equal to S21. We give new dependent choice‐based axiomatizations of the ∀trueΣ̂1normalb‐consequences of S21 and sans-serifR̂21. Our dependent choice axiomatizations give new normal forms for the normalΔ̂1b‐consequences of S21 and sans-serifR̂21. We use these axiomatizations to give an alternative proof of the finite axiomatizability of ∀trueΣ̂1normalbfalse(S21false) and to show new results such as ∀trueΣ̂1normalbfalse(R′31false) is finitely axiomatized and that there is a finitely axiomatized theory, sans-serifTUC, containing sans-serifŜ20 and contained in sans-serifR̂21. On the other hand, we show that our theory for ∀trueΣ̂1normalbfalse(sans-serifR̂21false) splits into a natural infinite hierarchy of theories. We give a diagonalization result that stems from our attempts to separate the hierarchy for ∀trueΣ̂1normalbfalse(sans-serifR̂21false).