Editing Saturated model (section)

==Examples==

Saturated models exist for certain theories and cardinalities:

* ('''Q''', <)—the set of [[rational number]]s with their usual ordering—is saturated.  Intuitively, this is because any type consistent with the [[dense linear order|theory]] is implied by the order type; that is, the order the variables come in tells you everything there is to know about their role in the structure.
* ('''R''', <)—the set of [[real number]]s with their usual ordering—is ''not'' saturated.  For example, take the type (in one variable ''x'') that contains the formula <math>\textstyle{x> -\frac{1}{n}}</math> for every natural number ''n'', as well as the formula <math>\textstyle{x<0}</math>.  This type uses ω different parameters from '''R'''.  Every finite subset of the type is realized on '''R''' by some real ''x'', so by compactness the type is consistent with the structure, but it is not realized, as that would imply an upper bound to the sequence −1/''n'' that is less than 0 (its least upper bound).  Thus ('''R''',<) is ''not'' ω<sub>1</sub>-saturated, and not saturated.  However, it ''is'' ω-saturated, for essentially the same reason as '''Q'''—every finite type is given by the order type, which if consistent, is always realized, because of the density of the order.
*A dense totally ordered set without endpoints is a [[eta set|η<sub>α</sub> set]] if and only if it is ℵ<sub>α</sub>-saturated.
* The [[countable random graph]], with the only non-logical symbol being the edge existence relation, is also saturated, because any complete type is isolated (implied) by the finite subgraph consisting of the variables and parameters used to define the type.

Both the theory of '''Q''' and the theory of the countable random graph can be shown to be [[categorical theory|ω-categorical]] through the [[back-and-forth method]].  This can be generalized as follows: the unique model of cardinality ''κ'' of a countable ''κ''-categorical theory is saturated.

However, the statement that every model has a saturated [[elementary extension]] is not provable in [[ZFC]].  In fact, this statement is equivalent to {{citation needed|date=July 2018}} the existence of a proper class of cardinals ''κ'' such that ''κ''<sup>&lt;''κ''</sup>&nbsp;=&nbsp;''κ''.  The latter identity is equivalent to {{nowrap|''κ'' {{=}} ''λ''<sup>+</sup> {{=}} 2<sup>''λ''</sup>}} for some ''λ'', or ''κ'' is [[strongly inaccessible]].