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UNIVERSAL CLASSES NEAR ${\aleph _1}$

Published online by Cambridge University Press:  21 December 2018

MARCOS MAZARI-ARMIDA  and
SEBASTIEN VASEY
MARCOS MAZARI-ARMIDA
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PENNSYLVANIA, USA E-mail: mmazaria@andrew.cmu.eduURL: http://www.math.cmu.edu/∼mmazaria/
SEBASTIEN VASEY
Affiliation:
DEPARTMENT OF MATHEMATICS HARVARD UNIVERSITY CAMBRIDGE, MASSACHUSETTS, USA E-mail: sebv@math.harvard.eduURL: http://math.harvard.edu/∼sebv/
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Abstract

Shelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:

Theorem.Assume${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.

  1. (1) If ψ is categorical in${\aleph _0}$and$1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) < 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.

  2. (2) If ψ is categorical in${\aleph _1}$, then ψ is categorical in all uncountable cardinals.

The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

Keywords

Primary 03C48Secondary 03C4503C5203C5503C75abstract elementary classesuniversal classescategoricitygood framestamenessprime models
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1633 - 1643
Copyright
Copyright © The Association for Symbolic Logic 2018 

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