On the structure of nonstandard models of arithmetic

Abstract:

In this paper we show that the additive group of each nonstandard model $^ \ast Z$ of the integers $Z$ is isomorphic to the group $\left \langle {F \times Z, + } \right \rangle$ where $F$ is a direct sum of $\alpha$-copies of the rational $Q,\alpha$ the cardinality of $^ \ast Z$, and + is defined by: $(a,x) + (b,y) = (a + b,x + y + g(a,b))$ for certain functions $g$ mapping from $F \times F$ to $Z$.