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A NOTE ON STANDARD SYSTEMS AND ULTRAFILTERS

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Abstract. Let (M, X) | = ACA0 be such that PX, 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. The standard system of a model M of PA (the first order formulation of Peano arithmetic) is the collection of standard parts of the parameter definable subsets of M, i.e., sets of the form X ∩ ω, where X is a parameter definable set of M, and ω is the set of natural numbers. It turns out that the standard system tells you a lot about the model; for example, any two countable recursively saturated models of the same completion of PA with the same standard system are isomorphic. A natural question to ask is then which collections of subsets of the natural numbers are standard systems. This problem has become known a

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