Sh:404

• Givant, S. R., & Shelah, S. (1994). Universal theories categorical in power and $\kappa$-generated models. Ann. Pure Appl. Logic, 69(1), 27–51. arXiv: math/9401213 DOI: 10.1016/0168-0072(94)90018-3 MR: 1301605
• Abstract:
  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 $T$.
• Version 1994-01-21_10 (22p) published version (25p)

@article{Sh:404,
 author = {Givant, Steven R. and Shelah, Saharon},
 title = {{Universal theories categorical in power and $\kappa$-generated models}},
 journal = {Ann. Pure Appl. Logic},
 fjournal = {Annals of Pure and Applied Logic},
 volume = {69},
 number = {1},
 year = {1994},
 pages = {27--51},
 issn = {0168-0072},
 mrnumber = {1301605},
 mrclass = {03C35 (03C45)},
 doi = {10.1016/0168-0072(94)90018-3},
 note = {\href{https://arxiv.org/abs/math/9401213}{arXiv: math/9401213}},
 arxiv_number = {math/9401213}
}