Universal theories categorical in power and kappa-generated models

Steven Givant; Saharon Shelah

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arXiv · math/9401213

Universal theories categorical in power and kappa-generated models

Steven Givant·Saharon Shelah

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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.

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Steven Givant, Saharon Shelah. 1994-01-15. Universal theories categorical in power and kappa-generated models. https://arxiv.org/abs/math/9401213

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