Elementary extension

An elementary extension is an extension preserving all first-order truth with parameters, hence giving elementary equivalence. An extension $N\supseteq M$ is elementary when every formula with parameters from $M$ has the same truth value in both structures.

A model $N$ is $\kappa$-saturated when every complete type over a subset of size less than $\kappa$ is realized in $N$.

A model is omega-saturated when it realizes every complete finite-tuple type over any finite parameter set. Realizing all one-variable types over finite parameter sets suffices by successively adjoining coordinates. This is weaker than saturation over every countable parameter set.

For every infinite regular cardinal $\kappa$, every first-order structure has a $\kappa$-saturated elementary extension. At one stage use the compactness theorem with the elementary diagram to realize all complete types over subsets of the stage having size less than $\kappa$. There is a set of such types; finite subsets of the enlarged theory are satisfiable because their finitely many witnesses are independent. Iterate for $\kappa$ stages, taking elementary unions at limits. Regularity puts any parameter set of size less than $\kappa$ inside one stage, and its type is realized at the next one. Passing to a larger regular cardinal covers singular requested degrees of saturation as well.

In a countable language, an ultraproduct over a nonprincipal ultrafilter on $\omega$ realizes every countable finitely satisfiable type, including types over countably many parameters. Enumerate the type ${\phi }_1,{\phi }_2,\dots$, and let $A_n$ be the ultrafilter-large set of coordinates where the first $n$ first-order formulas have a simultaneous witness. Put $B_n=\{i:i\ge n\}\cap {\bigcap }_{j\le n}{A}_j$. At coordinate $i$, choose a witness for the largest $n\le i$ with $i\in B_n$, or an arbitrary element when there is none. For each fixed $n$, the resulting function satisfies the first $n$ first-order formulas on $B_n$. The Łoś theorem then realizes the entire type. Countability of the language and parameter set makes the enumeration possible.

A $\kappa$-saturated model has the one-point extension property for partial elementary maps of size below $\kappa$: transport the type of the new element through the map and realize it by saturation. This is $\kappa$-homogeneity; back-and-forth gives the corresponding automorphisms when cardinalities permit.

If an aleph-zero-homogeneous model realizes every finite type over the empty set, it is aleph-zero-saturated. Realize the joint empty-set type of the parameters and a desired element, then use homogeneity to move the realized parameter tuple to the given one.

A model $N$ of cardinality $\kappa$ is universal when every model of the same complete theory having cardinality less than $\kappa$ admits an elementary embedding into $N$.

A model $N$ of cardinality $\kappa$ is homogeneous when every partial elementary map between subsets of size less than $\kappa$ extends to an automorphism of $N$.

A universal homogeneous model realizes every type over a small parameter set. Embed a small model containing a realization into the universal model, then use homogeneity to move the embedded parameters back to the original ones.

Let $N$ be uncountable and saturated, and let a formula define an equivalence relation. If every small elementary submodel of $N$ meets every class, then there are only finitely many classes. Otherwise a small elementary submodel $M$ yields the finitely satisfiable type $\{\neg E(x,m):m\in M\}$, contradicting saturation.

The union of an elementary chain of structures has a unique induced structure, and every member of the chain is an elementary substructure of the union.

A directed system of first-order structures and coherent elementary embeddings has a direct limit into which every member embeds elementarily. Represent an element by a pair consisting of a stage and an element at that stage; identify two representatives when their images agree at a common later stage. Interpret functions and relations at a common stage. Mathematical induction on first-order formulas proves elementarity: for an existential first-order formula true at the limit, its witness and parameters occur at a common stage, and elementarity pulls truth back to the stage containing the original parameters. This generalizes the elementary chain theorem beyond linearly ordered diagrams.

The complete type of a tuple over parameters records every formula over those parameters that the tuple satisfies.

A complete type is algebraic when it contains a formula having finitely many realizations in a model of the theory. A one-element example is the type containing $x=c$ for a named constant. Such a type cannot gain infinitely many realizations in an elementarily equivalent structure. An isolated type need not be algebraic: the unique empty-set one-type in a dense linear order without endpoints is isolated by $x=x$ and has infinitely many realizations.

With its standard topology, this type space is a Stone space. The type space $S_n^M(A)$ is the set of complete $n$-types over a parameter set $A$ that are consistent with the complete theory of $M$ together with its diagram over $A$.

For a type space $S_n^M(A)$, the basic open set $[\phi ]$ consists of complete types containing the formula $\phi$. These sets are clopen, satisfy $[\phi ]\cap [\psi ]=[\phi \wedge \psi ]$, and distinguish distinct types. The compactness theorem makes this topology compact; it is Hausdorff and has a basis of clopen sets.

A complete type $p\in S_n(A)$ is isolated when some formula $\phi \in p$ belongs to no other complete type in $S_n(A)$. Equivalently, the basic open set determined by $\phi$ is the singleton $\{p\}$.

If a complete countable theory has a prime model, isolated types are dense in every finite-arity type space. Every nonempty basic open set is realized in the prime model, whose finite tuples have isolated types.

A type $p(x_1,\dots ,x_n)$ is omitted in a structure $M$ when no tuple from $M^n$ realizes every formula in $p$.

Let $T$ be a consistent theory in a countable first-order language. For every countable family of nonisolated finite-arity types of $T$, there is a countable model of $T$ that omits every type in the family.

Let $p(\mathbf{x})$ be a nonprincipal partial type over a consistent first-order theory $T$. Suppose $T\cup \{\sigma (\mathbf{c})\}$ is consistent, with $\sigma$ a finite condition in finitely many new constants, and let $\mathbf{t}(\mathbf{c})$ be a tuple of closed terms. Some $\psi \in p$ leaves $T\cup \{\sigma (\mathbf{c}),\neg \psi (\mathbf{t}(\mathbf{c}))\}$ consistent. Otherwise the original-language first-order formulais consistent with $T$ and entails every member of $p$, contradicting nonprincipality. Replace all new constants occurring in the condition or terms by variables. This lemma makes the omission requirements dense in a countable Henkin construction.

Every complete type over parameters from $M$ is realized in some elementary extension of $M$. Apply compactness to the elementary diagram of $M$ together with a new tuple of constants satisfying the type.

Distinct complete types contain opposite choices $\phi$ and $\neg \phi$. The corresponding basic clopen sets partition the type space, so every type space is totally disconnected.

Quantifier elimination for dense linear orders shows that the one-types over $\mathbb{N}\subset \mathbb{Q}$ are determined by cuts: equality to a natural number, one of the intervals between consecutive natural numbers, the ray below zero, or the cut above every natural number. Only the last type is non-isolated.

A model is aleph-zero-homogeneous when every finite partial elementary map extends by one more element.