First- and Second-Order Models of Recursive Arithmetics

We study a quadruple of interrelated subexponential subsystems of arithmetic WKL 0 , RCA 0 , IΔ0, and ΔRA1, which complement the similarly related quadruple WKL0, RCA0, IΣ1, and PRA studied by Simpson, and the quadruple WKL 0 , RCA 0 , IΔ0(exp), and EFA studied by Simpson and Smith. We then explore the space of subexponential arithmetic theories between IΔ0 and IΔ0(exp). We introduce and study firstand second-order theories of recursive arithmeticARA1 and ARA2 capable of characterizing various computational complexity classes and based on function algebras A, studied by Clote and others.