On the existence of universal models

Archive for Mathematical Logic 43 (7):901-936 (2004)
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Abstract

Suppose that λ=λ <λ ≥ℵ0, and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of λ++ universal models of T of size λ+ for models of T of size ≤λ+, and is meaningful when 2λ +>λ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind possible applications in analysis, we further observe that for such λ, for any fixed μ>λ+ regular with μ=μλ+, it is consistent that 2λ=μ and there is no normed vector space over ℚ of size <μ which is universal for normed vector spaces over ℚ of dimension λ+ under the notion of embedding h which specifies (a,b) such that ||h(x)/||x∈(a,b) for all x

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Džamonja Mirna
University of East Anglia

Citations of this work

Universal Structures.Saharon Shelah - 2017 - Notre Dame Journal of Formal Logic 58 (2):159-177.
Club Guessing and the Universal Models.Mirna Džamonja - 2005 - Notre Dame Journal of Formal Logic 46 (3):283-300.

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References found in this work

Toward classifying unstable theories.Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (3):229-255.
Nonexistence of universal orders in many cardinals.Menachem Kojman & Saharon Shelah - 1992 - Journal of Symbolic Logic 57 (3):875-891.
Independence results.Saharon Shelah - 1980 - Journal of Symbolic Logic 45 (3):563-573.
On universal graphs without instances of CH.Saharon Shelah - 1984 - Annals of Pure and Applied Logic 26 (1):75-87.
Simple unstable theories.Saharon Shelah - 1980 - Annals of Mathematical Logic 19 (3):177.

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