Universal theories categorical in power and κ-generated models (original) (raw)

We investigate a notion called uniqueness in power κ that is akin to categoricity in power κ, 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 κ-unique for one uncountable κ, then it is κ-unique for every uncountable κ; in particular, it is categorical in powers greater than the cardinality of T .