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Examples of non-locality

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science M/C 249, University of Illinoisat Chicago, 851 S. Morgan Chicago, Illinois 60607, USA, E-mail: jbaldwin@uic.edu Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA, E-mail: shelah@math.huji.ac.il

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 (, ∞)-tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not (≤ ℵ2, ≤ ℵ2)-compact.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2008

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