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Preprint Number 1348

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1348. Marcos Mazari Armida and Sebastien Vasey
Universal classes near ℵ_1

Submission date: 7 December 2017


Shelah has provided sufficient conditions for an L_{ω_1, ω}-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 ℵ_n's. Using tools of Boney, Shelah, and the second author, we give assumptions on ℵ_0 and ℵ_1 which suffice when ψ is restricted to be universal:
Theorem. Assume 2^{ℵ_{0}} < 2 ^{ℵ_{1}}. Let ψ be a universal L_{ω_{1}, ω}-sentence.
- If ψ is categorical in ℵ_{0} and 1 ≤ I(ψ, ℵ_{1}) < 2 ^{ℵ_{1}}, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.
- If ψ is categorical in ℵ_1, then ψ is categorical in all uncountable cardinals.
The theorem generalizes to the framework of L_{ω_1, ω}-definable tame abstract elementary classes with primes.

Mathematics Subject Classification: Primary 03C48. Secondary: 03C45, 03C52, 03C55, 03C75

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Full text arXiv 1712.02880: pdf, ps.

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