Saturated Model - Motivation

Motivation

The seemingly more intuitive notion – that all complete types of the language are realized – turns out to be too weak (and is, appropriately, named weak saturation, which is the same as 1-saturation). The difference lies in the fact that many structures contain elements which are not definable (for example, any transcendental element of R is, by definition of the word, not definable in the field language). However, they still form a part of the structure, so we need types to describe relationships with them. Thus we allow sets of parameters from the structure in our definition of types. This argument allows us to discuss specific features of the model which we may otherwise miss – for example, a specific increasing sequence c_n having a bound can be expressed as realizing the type {x > c_n : n ∈ ω}, which uses countably many parameters. If the sequence is not definable, this fact about the structure cannot be described using the base language, so a weakly saturated structure may not bound the sequence, while an ω-saturated structure will.

The reason we only require parameter sets which are strictly smaller than the model is trivial: without this restriction, no infinite model is saturated. Consider a model M, and the type {x ≠ m : m ∈ M}. Each finite subset of this type is realized in the (infinite) model M, so by compactness it is consistent with M, but is trivially not realized. Any definition which is universally unsatisfied is useless; hence the restriction.