  We investigate a notion called uniqueness in power kappa that is akin to
categoricity in power kappa, but is based on the cardinality of the generating
sets of models instead of on the cardinality of their universes. The notion is
quite useful for formulating categoricity-like questions regarding powers below
the cardinality of a theory. We prove, for (uncountable) universal theories T,
that if T is kappa-unique for one uncountable kappa, then it is kappa-unique
for every uncountable kappa ; in particular, it is categorical in powers
greater than the cardinality of T.
