TY - JOUR
T1 - Universal theories categorical in power and κ-generated models
AU - Givant, Steven
AU - Shelah, Saharon
PY - 1994/9/6
Y1 - 1994/9/6
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=25844444938&partnerID=8YFLogxK
U2 - 10.1016/0168-0072(94)90018-3
DO - 10.1016/0168-0072(94)90018-3
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AN - SCOPUS:25844444938
SN - 0168-0072
VL - 69
SP - 27
EP - 51
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 1
ER -