### Universal theories categorical in power and $\kappa$-generated models

by Givant and Shelah. [GvSh:404]

Annals Pure and Applied Logic, 1994

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 .

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