Uncountable Real Closed Fields with PA Integer Parts

TL;DR

This paper explores the properties of uncountable real closed fields with integer parts related to nonstandard models of Peano Arithmetic, extending previous classifications to uncountable cases and analyzing real closures of specific models.

Contribution

It extends the classification of real closed fields with PA integer parts from countable to certain uncountable models, specifically focusing on -like models and their real closures.

Findings

Certain uncountable real closed fields with PA integer parts are characterized.

Some possibilities for extending previous countable classifications are ruled out.

Analysis of real closures of -like models provides new insights.

Abstract

D'Aquino, Knight and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of $\omega_1$-like models of PA.