Examples of non-locality
John T. Baldwin and Saharon Shelah
Source: J. Symbolic Logic Volume 73, Issue 3
(2008), 765-782.
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 (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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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1230396746
Digital Object Identifier: doi:10.2178/jsl/1230396746
Mathematical Reviews number (MathSciNet): MR2444267
Zentralblatt MATH identifier: 1162.03017
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