A note on standard systems and ultrafilters

TL;DR

This paper explores the relationship between Scott sets with definable ultrafilters and standard systems of models of arithmetic, showing how certain ultrafilters lead to models with specific properties.

Contribution

It demonstrates that Scott sets with definable ultrafilters coding consistent theories can be realized as standard systems of recursively saturated models of those theories.

Findings

Scott sets with definable ultrafilters can code models with specific standard systems.

Existence of end-extensions with prescribed standard systems.

Connection between ultrafilters and models of arithmetic.

Abstract

Let $(M,\scott X) \models \ACA$ be such that $P_\scott X$, the collection of all unbounded sets in $\scott X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\scott X$ such that $M$ thinks $T$ is consistent. We prove that there is an end-extension $N \models T$ of $M$ such that the subsets of $M$ coded in $N$ are precisely those in $\scott X$. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory $T$ extending first order arithmetic is the standard system of a recursively saturated model of $T$.