Models of Peano arithmetic as modules over initial segments

Abstract

Let $M$ be a countable non-standard model of first order Peano arithmetic (PA) and $I$ a weakly definable proper initial segment that is closed under addition, multiplication and factorial. We show that there is another model $N$ of PA such that the structure of $I$-module of $M$ coincides with that of $N$ and the multiplication of $M$ coincides with that of $N$ on $I$ but does not coincide at some $(a, b)\not\in I^{2}$.