The Formalization of The Arithmetic System on The Ground of The Atomic   Logic
  This paper is a continuation of the paper [16]. Namely, in [16] we have introduced, among others, the definition of the atomic entailment and we have constructed the system $\overset\sqcap S$, which is based on the atomic entailment. In this paper we show that the classical Arithmetic can be based on the Atomic Logic (see [17]). 
T.J. Stepien, L.T. Stepien
math.LO