A note on $\mathbb{Z}$ as a direct summand of nonstandard models of weak systems of arithmetic

TL;DR

This paper investigates the structural properties of nonstandard models of certain weak arithmetic systems, demonstrating that the additive group of these models cannot decompose with $bZ$ as a direct summand in some cases.

Contribution

It establishes the impossibility of $bZ$ being a direct summand in nonstandard models of $IE_2$, contrasting with models of $NOI$ where this is possible.

Findings

$bZ$ can be a direct summand in nonstandard models of $NOI$

$bZ$ cannot be a direct summand in nonstandard models of $IE_2$

Highlights differences between models of different weak arithmetic systems

Abstract

There are nonstandard models of normal open induction ($NOI$) for which $\mathbb{Z}$ is a direct summand of their additive group. We show that this is impossible for nonstandard models of $IE_2$.