### 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 .

Back to the list of publications