Source: J. Symbolic Logic
Volume 73, Issue 3
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 (2ℵ0,∞)-tame;
Theorem 3.3. It is consistent with ZFC that there is
an AEC with amalgamation which is not (≤ℵ2,≤ℵ2)-compact.
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