Past exam of the model theory course of the University of Oxford / 2011 / 3 / a

An omega-saturated model realizes every complete one-type over every finite subset of its domain that is consistent with the first-order structure's first-order theory with those named parameters. Equivalently, every partial type in a finite tuple over finitely many parameters that is finitely satisfiable in the first-order structure has a realization there. The equivalence follows by extending a partial type to a complete type; finite tuples can be handled one coordinate at a time.

For an example, take a countably infinite set in the language of equality alone. Over finitely many named elements, a complete one-type either requires equality with one of them or inequality with all of them. Both possibilities are realized. quantifier elimination for an infinite pure set follows by choosing a quantified variable either equal to one of the finitely many named values or distinct from all of them. Indeed every permutation fixing the parameters is a structure automorphism, and all other elements have the same type, so there are no further one-types. This first-order structure is omega-saturated.

For a counterexample, take $(\mathbb{N};0,S)$, where $S(n)=n+1$. The parameter-free partial typeis finitely satisfiable: choose an integer beyond the finitely many excluded numerals. It is not realized in the first-order structure, because every natural number is one of those numerals. Any complete extension of this partial type is therefore omitted as well. Thus the standard successor first-order structure is not omega-saturated.

Here fixed means fixed setwise: $\sigma [D]=D$ for every structure automorphism $\sigma$, not that each point must be fixed individually. A parameter-free definable set has this invariance by automorphism invariance of parameter-free definability.

For the converse, first establish the needed saturation implies homogeneity step. In a countable omega-saturated model, a finite partial elementary map can be extended to include any prescribed additional element in its domain. Transfer the element's complete type over the old finite domain along the map. Each finite part of this transported type has a witness because the old map is elementary; the omega-saturated model realizes the full transported type. The inequalities in the type maintain injectivity. Apply the same construction to the inverse map to enlarge the range. Alternating these steps through an enumeration of the countable domain is the back-and-forth method, giving a structure automorphism. In particular, any two elements with the same complete type over the empty set are related by a structure automorphism.

Assume now that $D$ is finite and invariant, and fix $a\in D$. All realizations in $A$ of $p_a=\mathrm{tp}(a/\varnothing )$ belong to $D$, by the preceding homogeneity and the invariance assumption. Suppose that no parameter-free first-order formula ${\theta }_a\in p_a$ has all its realizations in $D$. Then every finite subset ofis satisfiable in $A$: conjoin the finitely many first-order formulas from $p_a$, and use the assumed absence of a defining neighbourhood inside $D$ to find a witness outside $D$. This is a partial type over the finite parameter set $D$. The omega-saturated model would realize it by an element outside $D$ with the same empty-set type as $a$, a contradiction. Hence such a ${\theta }_a$ does exist.

There are only finitely many $a\in D$, soFor $D=\varnothing$, use $x\ne x$. Thus finite invariant sets in a countable saturated structure are definable. Finiteness is used twice: the extra parameters form a finite set, and the final disjunction is a first-order formula rather than an infinite disjunction.