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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Mathematics Subject Classification: 03E35, 03C55
In the list of publications of S. Shelah, this is publication number 614. Both authors thank the United States-Israel Binational Science Foundation for a partial support and various readers of the manuscript for their helpful comments.
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Džamonja, M., Shelah, S. On the existence of universal models. Arch. Math. Logic 43, 901–936 (2004). https://doi.org/10.1007/s00153-004-0235-1
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DOI: https://doi.org/10.1007/s00153-004-0235-1