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A downward Löwenheim-Skolem theorem for infinitary theories which have the unsuperstability property
Published online by Cambridge University Press: 12 March 2014
Abstract
We present a downward Löwenheim-Skolem theorem which transfers downward formulas from L∞,ω to 
, ω. The simplest instance is:
Theorem 1. Let λ > κ be infinite cardinals, and let L be a similarity type of cardinality κ at most. For every L-structure M of cardinality λ and every X ⊆ M there exists a model N ≺ M containing the set X of power ∣X∣ · κ such that for every pair of finite sequences a, b ∈ N
The following theorem is an application:
Theorem 2. Let λ<κ, T ∈ , ω, and suppose χ is a Ramsey cardinal greater than λ. If T has the (χ, , ω)-unsuperstability property, then T has the (χ, , ω)-unsuperstability property.
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- Research Article
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- Copyright © Association for Symbolic Logic 1988
Footnotes
1
I would like to thank John Baldwin for asking me a question which is the reason why I wrote this paper, and for his valuable remarks on the first draft. This research was partially supported by NSF grant DMS-8603167.
References
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