10 - Saturation

Summary

I have made numerous composites of various groups of convicts … The first set of portraits are those of criminals convicted of murder, manslaughter, or robbery accompanied with violence. It will be observed that the features of the composites are much better looking than those of the components. The special villainous irregularities have disappeared, and the common humanity that underlies them has prevailed.

Francis Galton, Inquiries into human faculty and its development (1907).

A saturated model of a complete theory T is a ‘most typical’ model of T. It has no avoidable asymmetries; unlike the man of Devizes, it's not short at one side and long at the other. (Or like the man of the Nore, it's the same shape behind as before.) We create saturated models by amalgamating together all possible models, rather in the spirit of Fraïssé's construction from section 7.1.

Although in a sense these are typical models, in another sense every saturated model A has some quite remarkable properties. Every small enough model of T is elementarily embeddable in A – this is called universality. Every type-preserving map between small subsets of A extends to an automorphism of A – this is strong homogeneity. We can expand A to a model of any theory consistent with Th(A) – this is resplendence.

These properties, particularly the resplendence and the strong homogeneity, make saturated models a valuable tool for proving facts about the theory T.