A note on standard systems and ultrafilters

Abstract

Let (M,𝒳) ⊨ ACA₀ be such that P_𝒳, the collection of all unbounded sets in 𝒳, admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in 𝒳 such that M thinks T is consistent. We prove that there is an end-extension N ⊨ T of M such that the subsets of M coded in N are precisely those in 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.