On the absoluteness of orbital ω-stability

We show that orbital ω-stability is upwards absolute for א0-presented abstract elementary classes satisfying amalgamation and the joint embedding property (each for countable models). We also show that amalgamation does not imply upwards absoluteness of orbital ω-stability by itself. Suppose that k = (K, k) is an abstract elementary class (or AEC; see [1, 8] for a definition), and let (M,a,N) and (P, b,Q) be such that M , N , P and Q are structures in Kא0 (where, for a cardinal κ, Kκ denotes the members of K of cardinality κ) with M k N , P k Q, a ∈ N \M and b ∈ Q \ P . The triples (M,a,N) and (P, b,Q) are said to be Galois equivalent or orbitally equivalent if M = P and there exist R ∈ Kא0 and k-embeddings π : N → R and σ : Q → R such that π and σ are the identity on M , and π(a) = σ(b). If k satisfies amalgamation (the property that if M , N and P are elements of K such that M k N and M k P then there exist Q ∈ K and k-embeddings π : N → Q and σ : P → Q such that π and σ are the identity on M ) then this relation is an equivalence relation on the class of such triples; each equivalence class is called a Galois type or orbital type (amalgamation is not necessary for orbital equivalence to be transitive). We say that the AEC k = (K, k) is ω-orbitally stable if, for each M ∈Kא0 , the set of equivalence classes overM as above for triples (M,a,N) withN ∈Kא0 is countable. An abstract elementary class k = (K, k) over a countable vocabulary τ is called א0-presentable (among other names, including PCא0 and analytically presented) if the class of models K and the class of pairs corresponding to k are each the set of reducts to τ of the models of an Lא1,א0 -sentence in some expanded language (this formulation implies that the Löwenheim-Skolem number of k is א0, which in any case we take to be part of the definition). Equivalently (assuming that the Löwenheim-Skolem number of k is א0), k is א0-presentable if the collections of subsets of ω coding (in some natural fashion) the restrictions of K and k to countable structures are analytic. If K is א0-presentable, the ω-orbital stability of K is naturally expressed as a Π4 property in a countable parameter for K. One might hope that this property has a simpler definition, and moreover that the property is absolute between models of set theory with the same ordinals. In this note we show that ω-orbital stability is upwards absolute for א0-presentable abstract elementary classes k = (K, k) for which (Kא0 , k) satisfies amalgamation and the joint embedding property (the property that any two elements of K can be k-embedded in a common element of K, i.e., that (K, k) is directed). We also present an א0-presented AEC, satisfying amalgamation but not the joint embedding ∗Supported in part by NSF Grant DMS-1201494. †Research partially support by NSF grant DMS 1101597. Publication number 1073 on Shelah’s list.