An extendible model with a rigid elementary extension

TL;DR

This paper constructs an example of a countable model that is extendible, meaning it shares its Scott sentence with an uncountable structure, and also has a rigid elementary extension, challenging previous assumptions.

Contribution

It provides the first known example of an extendible model possessing a rigid elementary extension, highlighting new complexities in model theory.

Findings

Existence of an extendible model with a rigid elementary extension

Counterexample to previous beliefs about rigidity and extendibility

Advances understanding of Scott sentences and model extensions

Abstract

A countable structure is said to be extendible if it has the same Scott sentence as some uncountable structure. Rigid structures are not extendible. We give an example of an extendible model with a rigid elementary extension.