On the Value Group of a Model of Peano Arithmetic

TL;DR

This paper explores the structure of value groups in $IPA$-real closed fields, revealing their exponential group nature, and classifies these groups for countable recursively saturated cases, also constructing examples outside this class.

Contribution

It characterizes the value groups of $IPA$-real closed fields as exponential groups and provides a classification for countable recursively saturated cases, highlighting the limitations of the converse.

Findings

Value group of an $IPA$-real closed field is an exponential group.

The converse of the main characterization does not hold in general.

Constructs countable exponential real closed fields that are not $IPA$-real closed fields.

Abstract

We investigate $IPA$ - real closed fields, that is, real closed fields which admit an integer part whose non-negative cone is a model of Peano Arithmetic. We show that the value group of an $IPA$ - real closed field is an exponential group in the residue field, and that the converse fails in general. As an application, we classify (up to isomorphism) value groups of countable recursively saturated exponential real closed fields. We exploit this characterization to construct countable exponential real closed fields which are not $IPA$ - real closed fields.