A note on standard systems and ultrafilters

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 as the Scott set problem. For countable models the standard systems are exactly the countable Scott sets, i.e., countable boolean algebras of sets of natural numbers closed under relative recursion and a weak form of König’s lemma [Scott, 1962]. It follows, by a union of chains argument (see [Knight and Nadel, 1982]), that for models of cardinality at most א1 the standard systems are exactly the Scott sets of cardinality at most א1. If the continuum hypothesis holds this settles the Scott set problem. However, if CH fails then very little is known about standard systems of models of cardinality strictly greater than א1, although it is easy to see that any standard system of any model is a Scott set (see [Kaye, 1991]). The key notion of a definable ultrafilter plays a central role in this paper. Suppose X is a collection of sets of natural numbers, and PX is the collection of infinite members of X . A filter F on PX is said to be definable if for all A ∈ X { a ∈ ω | (A)a ∈ F } ∈ X , where (A)a = { b ∈ ω | 〈a, b〉 ∈ A } and 〈·, ·〉 is some canonical pairing function. The central theorem of this paper is Theorem 1 below (whose proof will be presented in section 2). Theorem 1. Let X be a Scott set that carries a definable ultrafilter and let T ∈ X be a consistent completion of PA. Then there is a recursively saturated model of T with standard system X .