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A note on standard systems and ultrafilters

Journal of Symbolic Logic
Year of publication: 
Scientific journal article - peer reviewed

Let (M,X) |= ACA_0 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 .

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