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Applications of vaught sentences and the covering theorem1

Published online by Cambridge University Press:  12 March 2014

Victor Harnik  and
Michael Makkai
Victor Harnik
Affiliation:
Dartmouth College, Hanover, New Hampshire 03755
Michael Makkai
Affiliation:
Université de Montréal, Montréal, Québec, Canada
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Abstract

We use a fundamental theorem of Vaught, called the covering theorem in [V] (cf. theorem 0.1 below) as well as a generalization of it (cf. Theorem 0.1* below) to derive several known and a few new results related to the logic . Among others, we prove that if every countable model in a PCφ1ω class has only countably many automorphisms, then the class has either ≤ℵ0 or exactly nonisomorphic countable members (cf. Theorem 4.3*) and that the class of countable saturated structures of a sufficiently large countable similarity type is not among countable structures (cf. Theorem 5.2). We also give a simple proof of the Lachlan-Sacks theorem on bounds of Morley ranks (§7).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 1 , March 1976 , pp. 171 - 187
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

The authors' work was supported by a grant of the National Research Council of Canada.

References

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