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ALMOST GALOIS ω-STABLE CLASSES

Published online by Cambridge University Press:  22 July 2015

JOHN T. BALDWIN
Affiliation:
DEPARTMENT OF MATHEMATICS STATISTICS AND COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO USAE-mail: jbaldwin@uic.edu
PAUL B. LARSON
Affiliation:
DEPARTMENT OF MATHEMATICS MIAMI UNIVERSITY OXFORD, OHIO, USAE-mail: larsonpb@miamioh.edu
SAHARON SHELAH
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERSUSALEM JERUSALEM 91904 ISRAEL and DEPARTMENT OF MATHEMATICS RUTGERS, THE STATE UNIVERSITY OF NEW JERSEY PISCATAWAY, NEW JERSEY 08854-8019, USAE-mail: shelah@math.huji.ac.il

Abstract

Theorem. Suppose that k = (K, $$\prec_k$$) is an 0-presentable abstract elementary class with Löwenheim–Skolem number 0, satisfying the joint embedding and amalgamation properties in 0. If K has only countably many models in 1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$) is an 0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality 1. The assertion that K is 1-categorical is then absolute.

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Copyright
Copyright © The Association for Symbolic Logic 2015 

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