Editing Saturated model (section)

==Relationship to prime models==

The notion of saturated model is dual to the notion of [[prime model]] in the following way: let ''T'' be a countable theory in a first-order language (that is, a set of mutually consistent sentences in that language) and let ''P'' be a prime model of ''T''. Then ''P'' admits an [[elementary embedding]] into any other model of ''T''. The equivalent notion for saturated models is that any "reasonably small" model of ''T'' is elementarily embedded in a saturated model, where "reasonably small" means cardinality no larger than that of the model in which it is to be embedded.  Any saturated model is also [[homogeneous model|homogeneous]].  However, while for countable theories there is a unique prime model, saturated models are necessarily specific to a particular cardinality.  Given certain set-theoretic assumptions, saturated models (albeit of very large cardinality) exist for arbitrary theories.  For ''λ''-[[stable theory|stable]] theories, saturated models of cardinality ''λ'' exist.