Examples of non-locality

Abstract

We use κ-free but not Whitehead Abelian groups to construct Abstract Elementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC’s which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is an AEC with amalgamation which is not (ℵ₀,ℵ₁)-tame but is (2^ℵ₀,∞)-tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not (≤ℵ₂,≤ℵ₂)-compact.