1 results
UNIVERSAL CLASSES NEAR
${\aleph _1}$
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 4 / December 2018
- Published online by Cambridge University Press:
- 21 December 2018, pp. 1633-1643
- Print publication:
- December 2018
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- Article
- Export citation
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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) 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) 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.
