Saturated Model - Examples

Examples

Saturated models exist for certain theories and cardinalities:

• (Q, <) – the set of rational numbers with their usual ordering – is saturated. Intuitively, this is because any type consistent with the 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 numbers with their usual ordering – is not saturated. For example, take the type (in one variable x) which contains the formula for every natural number n, as well as the formula . 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 it is consistent with the structure, but it is not realized, as it would imply an upper bound to the sequence −1/n which is less than 0 (its least upper bound). Thus (R,<) is not ω₁-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.
• The countable random graph, with the only non-logical symbol being the edge existence relation, is also saturated, because any complete type is implied by the finite subgraph consisting of the variables and parameters used to define the type.

Both of these theories can be shown to be 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 the existence of a proper class of cardinals κ such that κ<κ = κ. The latter identity implies that either κ = λ+ = 2λ for some λ, or κ is weakly inaccessible.