Value Groups of Real Closed Fields and Fragments of Peano Arithmetic

Merlin Carl, Paola D'Aquino, Salma Kuhlmann

Published 2012-05-10, updated 2014-11-24Version 5

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. This result continues to hold for weaker fragments of Peano Arithmetic, such as $I\Delta_{0}+EXP$. As an application, we construct a class of real closed fields which are not IPA. 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.