Models of expansions of N with no end extensions

TL;DR

This paper constructs models of Peano arithmetic with no elementary end extensions using creature forcing, revealing a Borel uncountable set of subsets that achieve this property and identifying arithmetically closed sets lacking definably closed ultrafilters.

Contribution

It introduces a novel method using creature forcing to produce models of PA with no end extensions and characterizes sets with specific ultrafilter properties.

Findings

Existence of models of PA with no elementary end extensions

A Borel uncountable set of subsets of N suffices for such expansions

Identification of arithmetically closed sets without definably closed ultrafilters

Abstract

We deal with models of Peano arithmetic (specifically with a question of Ali Enayat). The methods are from creature forcing. We find an expansion of N such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of N such that expanding N by any uncountably many of them suffice. Also we find arithmetically closed A with no definably closed ultrafilter on it.