  We show that the class of 1-exact operator systems is not uniformly definable
by a sequence of types. We use this fact to show that there is no finitary
version of Arveson's extension theorem. Next, we show that WEP is equivalent to
a certain notion of existential closedness for C$^*$ algebras and use this
equivalence to give a simpler proof of Kavruk's result that WEP is equivalent
to the complete tight Riesz interpolation property. We then introduce a variant
of the space of n-dimensional operator systems and connect this new space to
the Kirchberg Embedding Problem, which asks whether every C$^*$ algebra embeds
into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. We end with some
results concerning the question of whether or not the local lifting property
(in the sense of Kirchberg) is uniformly definable by a sequence of types in
the language of C$^*$ algebras.
