A note on standard systems and ultrafilters

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 $T$.

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