On the existence of universal models

by Dzamonja and Shelah. [DjSh:614]
Archive for Math Logic, 2004
Suppose that lambda = lambda^{< lambda} >= aleph_0, and we are considering a theory T . We give a criterion on T which is sufficient for the consistent existence of lambda^{++} universal models of T of size lambda^+ for models of T of size <= lambda^+, and is meaningful when 2^{lambda^+}> lambda^{++} . 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 aplpications in analysis, we further observe that for such lambda, for any fixed mu > lambda^+ regular with mu = mu^{lambda^+}, it is consistent that 2^lambda = mu and there is no normed vector space over Q of size < mu which is universal for normed vector spaces over Q of dimension lambda^+ under the notion of embedding h which specifies (a,b) such that |h(x) |/ |x | in (a,b) for all x .

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