Vaught's Two-Cardinal Theorem and Quasi-Minimality in Continuous Logic

2017/10/16 by Noquez, Victoria
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1710.05809

Abstract

We prove the following continuous analogue of Vaught's Two-Cardinal Theorem: if for some κ>λ≥ ℵ₀, a continuous theory T has a model with density character κ which has a definable subset of density character λ, then T has a model with density character ℵ₁ which has a separable definable subset. We also show that if we assume that T is ω-stable, then if T has a model of density character ℵ₁ with a separable definable set, then for any uncountable κ we can find a model of T with density character κ which has a separable definable subset. In order to prove this, we develop an approximate notion of quasi-minimality for the continuous setting. We apply these results to show a continuous version of the forward direction of the Baldwin-Lachlan characterization of uncountable categoricity: if a continuous theory T is uncountably categorical, then T is ω-stable and has no Vaughtian pairs.

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