Cardinality
In contrast with the notion of saturated model, prime models are restricted to very specific cardinalities by the Löwenheim-Skolem theorem. If is a first-order language with cardinality and a complete theory over then this theorem guarantees a model for of cardinality therefore no prime model of can have larger cardinality since at the very least it must be elementarily embedded in such a model. This still leaves much ambiguity in the actual cardinality. In the case of countable languages, all prime models are at most countable.
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